Imagine_Maths_CB_Grade7_Production_Sample_Part1

MATHEMATICS Master Mathematical Thinking

Grade 7

Fo re wo rd

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

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

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

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

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

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

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

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

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

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

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

K ey El ements o f a C hDivision apt easr— a Q u i c k G lanc e Repeated Subtraction

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

Komal has 20 coins, and she wants to put 5 coins in each pouch. Let us help her to divide 20 by 5 using repeated subtraction.

Th

H

T

O

2

5

4

8

4 Tens

8 Ones

2 Thousands 5 Hundreds

Step 1 Subtract 5 from 20 until we get 0.

2

0

Total number of coins = 20.

Understanding Multiplication – Let's Warm-up Introductory Concept Write the correct place value of the coloured numbers.

Let’s take away 5.

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

1 32 Together Number of remaining coins = 15.

– ays is always fun!quick You first decide on a place to visit, warm-up with a real life 2 548 you want__________ 1 2 Let’s take away 5 again. what you can do there, and then finally make arrangements.

1 3 876 example __________ Number of remaining coins = 10.

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

– Let’s take away 5 again.

5 9958 __________ Real Life Connect

5

He starts adding quickly.

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

He has learnt 14 words already! Hurray!

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

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

0 1 Friday 2 3 Thursday

30

31

6

1

6×1=6

7

13

8

14

20

15

3

6 × 2 = ____

9 16

TTh Th H T O Th Number of remaining coins =H 0. T +

28

29

3

4

4

=31 ×

5

multiplicand

= 62

Th

5

H

3

T

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

5

product

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

Let us multiply 123 and 3.

6

Find the factors of 36 using the division method.

Divide numbers by 36

What do we get?

Check the remainder

Are the numbers factors of 36?

36 ÷ 3

12

____

____

0

____

Step 1

Step 2

Step 3

Multiply by ones.

Multiply by tens.

Multiply by hundreds.

Multiply 3 and 3 ones.

Multiply 3 and 2 tens.

Did You Know?

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

of books step by step.

What do we know?

The total number of non-fiction books = 1567 Subtract 7 from 21 until we get 0.

What do we need to find?

1

dividing 92 by weby get 2 remainder.

On dividing 96 by 6, we get ____ remainder.

36 ÷ 1 36 the number 0 of a number areOn the products we6,get multiplying by 1,Yes2, Think = The total number of fiction books 1 – and 7 Tell 36 ÷ 2 ____ 0 ____ The total number of non-fiction books Do we need+of to go6.

So, 92 _______________________________ of 6.

So, 96 ___________________________ The total number of books in the library

1

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Step 2 Count how many times 7 has been subtracted.

Th

H

T

O

1

5

6

7

We subtracted 1 2 1 9 seven 3 times. +

So, there are three 7s in 21.

2 7 8 6

4

beyond 6=to1219 find + 1567 more factors of 36?

49

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1 total number of fiction books = 1219 96StepThe 2

123 × 3 = 369

123 × 3 = __ __9

How many 7s are there in 21?

Example 2

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

6 92

4

0 O

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

So, there are four 5s in 20. Hence, Komal requires 4 pouches to keep the coins.

Let us start finding the total number

even number?

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

TTh

2

multiplier

–

We can multiply two numbers in any order.

The product always remains the same.

Horizontal Method

9/11/2023 4:24:58 PM books and 1567 non-fiction books. How many books are there in total?

31

ultiples

O

1

We subtracted 5, four times.

In the library, there are 1219 fiction to school the concept

all the circled numbers

Are the multiples of an even number always an

Do It Together

1

Story Sums

Think and Tellhave in common?

30

1

3

5

Count Fun the number of times 5 has been subtracted. fact, related

What do you think do

UM24CB_G4.indb 1

27

2

5

Multiplication Rules

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

Do It Together

1

24

We know that the number obtained from multiplication is the product.

Add 12,344 and 1115.

6 × ____ = ____

StepSo, 2 of 5. + 1115 = _____________.

Thinkofand Tell I scored out12,344 The first five multiples 6 are 6, ____, ____, 24, ____._________ 17

23

Multiplication by 1-digit Number

Let’s take away 5 again.

6 × ____ = 24

10

A quick-thinking

21 question 22

6 × ____ = ____

2

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

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

2

5 0

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

Wednesday

1

5

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

Solve to find the answer.

– 7 out 36 ÷ 5 7 1 ____ 3 × 6 = An 18important So,Pointing the total number of books in the library is 2786.

36 ÷ 6 ____ 0 Yes It Yourself 5A 7 keep in commonly made 3 × 7 = point 21 toDo Error Alert!

The city NGO organised a two-day donation drive.

Remember!

Remember! the first day of the drive, 1366 clothes were So, the factors of 36 are ___________________________________________________________________.

3 × 8 = mind 24 3 mistakes and – On 7 we add 1000 to a collected. On the second day of the drive, 1000 Always countWhen how many times the A number is a multiple of itself too.

4-digit number, only the clothes were collected. How many clothes were 3 × 9 = 1 27Colour the balloons that are multiples of 2. avoid number has digit been not in thesubtracted, thousands 0howinto For example, multiples of 5 are 5, collected total? place changes.

Do15, It Yourself the number itself.

3 × 10 = 30 10, 20, 25,5C 30 and so on! them 2

4

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

To the nearest 10s

Example 4

To the nearest 100s

a 1147 ÷ 2

b 4589 ÷ 3 c 6478 ÷ 6

d 8974 ÷ 7

1

2

3

1

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

2

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

4

Find the factors of the following numbers using multiplication.

What do we know?

9

ainder

15 – 15 00

3 11

5 d 3916 e 340 14 16 18 15 13 – 17 15 Find the factors of the following numbers using division.

5 a b c d e 9 11 12 13 15 leaves remainder 1 01 a 14

12

HOTS:

b 21

c 36

1 238 children went to a school camp. If one tent can be shared by 4 children, Total number of clothes collected = 1366 + 1000

g 48

f 12 questions k 17

b 8

g 13 l

18

c 9

d 10

h 14

i

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9

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

10

What is the smallest number that has exactly three factors?

s and Factors

Points to Remember m

Word Problems

19

15

n 20

The place value table is divided into groups called periods.

•

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

•

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

ways can she arrange the eggs?

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92

A

and fun

Chapter end

9/11/2023 4:25:17 PM

summary

e 11 j

16

83

QR Code: Access

Place value Scavenger Hunt:

Divide the entire class into groups of 4.

classroom

2

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

activity

3

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

4

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

5

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

The number being divided is called the dividend. The number by which we divide is called the divisor. The result of the division is called the quotient. The number left over after division is called the remainder.

•

To check if our answer after division is correct, we can use: Dividend = (Quotient × Divisor) + Remainder.

•

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

•

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

•

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

•

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

•

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

•

When a number is divided by 1000, the digits in the ones place, tens place and thousands place form the remainder and the remaining digits form the quotient.

77

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to interactive

Setting: Groups of 4 1

•

Chapter 4 • Division

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

Materials Required: Newspapers, Magazines, or the Internet

UM24CB_G4.indb 92

Estimate the amount of money that each student gets.

15-09-2023 3.0

of biscuits each packet. are possible?

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

Math Lab

multidisciplinary

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

Points to Remember

Raman, a baker, baked biscuits.

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

ers up to ts •

2

o 25

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

•

approximately how many tents will be needed for the camp?

77

81

•

•

i

20 30

18

by 5, we get 0 remainder.

On dividing 5,Explain we get remainder 1.

Is of 6 a 16 factorby of 64? your answer.

7 Find the first five multiples the given numbers.

2 Applicative iple of 5.

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

Which a numbers betweenof 1 and a 7 and analytical

h 50

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Is 18 a factor of 126? Explain your answer.

6

115

Word Problems

What do we need to find?

19 f

people. About how many tables are needed?

Chapter 6 • Division

42

f

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

10

[Round off the dividend to the nearest hundred].

Number of clothes collected during the second day of the drive = 1000

eck if a number is a multiple of a number using division. If the remainder Show 20 in different arrangements. Then, list the factors of 20. gger number is a multiple of the other3number.

For example: 5

e 5555 ÷ 5

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

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

Place value and face value of each digit.

Correct number representation.

Correctly written number names.

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

iii

22

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

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

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

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

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

Unit Component

Snapshot

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

Real Life Connect

Father: Thank you.

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

Father: Yes.

Real Life Connect

Ajay: But, what is this big number 781005?

From: Ajay Shukla, 12, Hathipol,

Guwahati - 781005

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

Theoretical explanation

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

Facts about Multiples

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

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

All Numbers!

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

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

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

I do

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

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

Find the 5 multiples of 4.

9999 + 1first = 10000

Remember!

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

10000 is read as “Ten Thousand”.

1

2

3

4

Let us learn more about 5-digit numbers!

0

1

2

3

4

5

6

7

8

5

number. 99999 is the greatest 5-digit number.

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

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

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

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

4 × 3 = 12

Examples

multiples of 4. For example

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

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

Example 2

Did You Know?

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

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 its0position in the given number.

The place 100 5= ×400.

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

iv

2

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

5

UM24CB_G4.indb 2

5 × 5 = 25

leaves remainder 0

95 – 95 00

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

9/11/2023 4:24:58 PM

Find the factors of 36 using the division method.

Do It Together

GRR Steps

Divide numbers by 36

Unit Component

What do we get?

Check the remainder

36 ÷ 1

36

0

36 ÷ 2

____

0

____

Yes

36 ÷ 4

____

0

____

Think and Tell Do we need to go

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

Do It Together

Find the method.

Step 1 factors7of 36 using1 the division 36 ÷ 5 ____

Do It Together

Factors of 20 36 ÷ 6 Divide numbers by 36

We do

Are the numbers

factors of 36?

Snapshot

1 2 0 What do we get?

____

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

Step 2 So, the factors of 36 are ___________________________________________________________________.

Do It Together

36 ÷of130 Factors

36

0

Yes

Step363÷ 2

____

0

____

Common 36 ÷ 3Factors

12

____

____

Do It Yourself 5C

1

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

2

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

7

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

The common factors of 20 and 30 are ____________________________________________________.

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

3 4

Think and Tell

36 ÷ 6

____

0

Yes

Math Lab of the following Find the factorscommon numbers multiplication.

The highest factor of 20 using and 30 is __________.

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

Board Game of Multiples

Find the factors of the following numbers using division.

Do It Yourself 5D

5

Setting:bIn groupsc of 4

a 9

11

12

d 13

e 15

f

18

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

Find the common factorsyour of the following numbers.

1 18 Is a factor of 126? Explain answer.

6

8, 10 of 64? Explainbyour 12, answer.

15 IsMethod: 6a a factor

7

c 13, 16

d 14, 20

e 16, 18

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

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

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

31

number.

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

3 Find thecommon factors of the following using multiplication.

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

Do It Yourself

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

Word Problems

1

a 16 and 24

1

16

a common factor of allyour the numbers.

Isc 6 0aisfactor of 64?

Explain answer. _______

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

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

5

9/11/2023 4:26:15 PM

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

Word Find theProblems factors.

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

ways can she arrange the eggs?

e 98 95 j

222

Find the common factors of the given pairs of numbers.

2

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

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

92

a 5 and 20

b 20 and 100

c 12 and 36

d 60 and 200

c 15

d 23

Write the first 5 multiples. a 7

b 11

e 30

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

UM24CB_G4.indb 92

a 2 and 5

e 10 and 15

6

77

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

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

24 and 30 What different and 12 d 20 and 25

5

i

42

7

UM24CB_G4.indb 95

4

9

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

f

3

6 10

aChapter 45 5 • Multiples andbFactors 66 2

14

13 have no common _______ aand factor ofwho 126?

Explain factors. yourmost answer.

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

5Isb 1811

9

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79

f 818

6

92 8

Chapter Checkup

33

h 50 11 60

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

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

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

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

You do

28

18 inhas different arrangements.

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

Circle correct 9 2 Which 5 18.

24 option.

15 Verify 42your answer.

25 35 2 2Show

b 3 and 7 f

10 and 25

c 5 and 8

g 11 and 22

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d 4 and 14

h 20 and 24

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

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

b Multiples of 6 that are smaller than 50.

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

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

97

Chapter 5 • Multiples and Factors

v

Gradual Release of Responsibility UM24CB_G4.indb 97

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C o nt e nt s

1 2 3 4 5 6 7

vi

Operations on Fractions .....................................................................................................1 Operations on Decimals ...................................................................................................17 Integers ..............................................................................................................................30 Measures of Central Tendency ........................................................................................45 Data Handling ....................................................................................................................64 Introduction to Probability ..............................................................................................89 Simple Equation...............................................................................................................102

1

Operations on Fractions

Let’s Recall A fraction represents a part of a whole. It has two parts: a numerator and a denominator.

Let’s say Raghav has a chocolate bar, and he divides it into 5 equal parts. So, here, a chocolate bar is considered a whole, and each part is considered part of a whole. He ate 3 parts out of 5 equal parts.

To represent this as a fraction, Raghav shades 3 parts out of 5.

3 The shaded part represents .

5

Here, 3 is the numerator and 5 is the denominator.

Types of fractions

Examples

A proper fraction is a fraction where the numerator is smaller than the denominator. 1 3 5 7 , , , A proper fraction is always less than 1.

2 4 9 8 An improper fraction is a fraction where the numerator is equal to or greater than the denominator. An improper fraction is always greater than or equal to 1.

5 7 11 13 5 , , , , 2 4 9 8 5

3 3 1 5 A mixed fraction is a combination of a whole number and a proper fraction. A mixed 2 ,5 ,7 ,6 4 4 9 11 fraction is another way of representing an improper fraction. It is always greater than 1.

Let’s Warm-up Fill in the blanks.

1 The numerator is always _________ than the denominator for proper fractions.

2 A mixed fraction has a __________ part and a fractional part.

3 __________ is the denominator in 5.

8 17 4 The numerator in is __________.

19 5 An improper fraction cannot be __________ than 1.

I Scored __________ out of 5.

Mean, Median and Mode of Fractions Addition and Subtraction Real Life Connect

2 Riya is happy to get her first salary. She spends of her salary on 5 1 house rent, salary on food and saves the remaining salary.

4

Adding and Subtracting Fractions Riya wanted to check what fraction of her salary she spent.

How could she do that?

Let us find out!

To find the fraction of salary spent, we will add the individual parts of the salaries spent.

Addition of Fractions

2 Fraction of salary spent by Riya on house rent = 5 1 Fraction of salary spent by Riya on food = 4 2 1 Fraction of total salary spent by Riya = + 5 4 Take LCM of 5 and 4 LCM of 5 and 4 is 20.

2 2 4 8 = × = 5 5 4 20

Remember!

Sum of like fractions =

1 1 5 5 = × = 4 4 5 20

2 1 8 5 8 + 5 13 + = + = = 5 4 20 20 20 20 13 Thus, Riya spends of her salary.

20 Addition of mixed numbers 1 1 Add 4 + 2 5 3 Whole number part → 4 + 2 = 6 1 1 1 3 1 5 3 5 8 Fractional part → + = × + × = + = 5 3 5 3 3 5 15 15 15 8 8 6+ =6 15 15

Did You Know?

Musicians use fractions to understand and create rhythms.

1 1 8 =2 =6 5 3 15 Properties of Addition of Fractions Thus, 4

Commutative Law

2

Sum of numerators Common denominator

Associative Law

a c c a + = + b d d b

a c e a c e + + + = + b d f b d f

1 1 3 2 (3 + 2) 5 + = + = = 2 3 6 6 6 6

1 1 1 3 2 1 5 1 10 3 13 + + + = + = + = + = 2 3 4 6 6 4 6 4 12 12 12

1 1 2 3 2+3 5 + = + = = 3 2 6 6 6 6

1 1 1 1 4 3 1 7 6 7 13 + + + = + = = + = + 2 3 4 2 12 12 2 12 12 12 12

Subtraction of Fractions

13 of her salary. What fraction of her salary is left with Riya?

20 Fraction of salary left = Total salary – Fraction of salary spent 13 =1− 20 20 13 = − Remember!

20 20 Difference of numerators 7 Difference of like fractions = = Common denominator 20 Subtraction of Mixed numbers 4 1 Subtract 9 – 3 5 6 Covert both the fractions into improper fractions.

We know that Riya spent

4 49 9 = 5 5

1 19 3 = 6 6

LCM of 5 and 6 = 30 49 294 = 5 30

19 95 = 6 30

Take LCM of 5 and 6.

4 1 294 95 199 19 9 –3 = − = =6 5 6 30 30 30 30

Think and Tell Check if the associative and commutative properties hold true or not for the subtraction of fractions.

4 1 19 Thus, 9 – 3 = 6 5 6 30

Example 1

Add. 8 5 a + 9 6

8 8 × 2 16 = = 9 9 × 2 18 16 15 31 + = 18 18 18

5 5 × 3 15 = = 6 6 × 3 18

31 (1 × 18) + 13 18 13 13 = = + =1 18 18 18 18 18 Thus,

Example 2

5 1 +1 9 2 Whole number part → 2 + 1 = 3

b 2

8 5 13 = =1 9 6 18

Fractional part →

5 1 5 2 1 9 10 9 19 + = × + × = + = 9 2 9 2 2 9 18 18 18

5 1 10 + 9 2 + 1 = (2 + 1) + 9 2 18 =3+ Thus, 2

19 1 1 =3+1 =4 18 18 18

5 1 1 +1 =4 9 2 18

1 7 Parth jogged 3 km and Priya jogged km. Who jogged a longer distance? How much more?

3 8 1 7 Distance jogged by Parth = 3 km; Distance jogged by Priya = km 3 8 7 1 To find how much more Parth jogged, subtract km from 3 km.

8 3 1 7 10 7 =3 − = − 3 8 3 8 10 8 7 3 80 21 59 11 LCM of 3 and 8 = 24 = × − × = − = =2 3 8 8 3 24 24 24 24 Thus, Parth jogged 2

11 km more than Priya.

24

Chapter 1 • Operations on Fractions

3

Do It Together

1 1 1 Add: 2 + + 5 2 5 4 (2 + 5) +

7 3 Subtract: 5 − 2 8 7

1 1 1 + + 2 5 4

5

7 47 = 8 8

3 17 2 = 7 3

Do It Yourself 2A 1

Solve. a

2

4

5

9 + 2 22 11

b

c

7 2 − 13 9

d

2 − 30

e

3 +5 40 6

c

2 + 1 − 3 17 17 17

d

5 + 1 − 3 12 12 12

e

15 −

b

1

d

3 0 + 50 = 8

Simplify the below fractions. a

3

2 5 + 3 9 2 + 1 + 3 17 17 17

b

3

4 + 12 + 21 9 9 9

Fill in the blanks.

9 9 + ________ = 2 11 11

a

2

c

11 21 21 + = + 19 40 40

Compare using appropriate symbols. (>, <, =)

2 1 + + 4 = 12 + 3 6 5 3

a

2

2 1 2 1 + 3 _____ 7 − 2 3 5 4 3

b

1

c

8

5 1 7 − 4 _____ 3 7 4 8

d

11 + 1 1 _____ 23 + 1 9 2 7 7

73

7 +45 − 3 8 6 12

+4 5

5 2 1 3 − _____ + 8 3 2 4

Answer the given questions.

2 7 a How much is 5 less than 7 ?

3 8 2 1 b What must be added to 2 to get 4 ?

5 4 c

d e

1 8 4 cm, 1 cm, cm.

4 9 5 7 2 cm long and cm wide.

Find the perimeter of a rectangle which is 11 9 1 2 3 5 Subtract the difference of 7 and 9 from the sum of 2 and 4 .

4 5 4 6 What is the perimeter of a triangle having its sides as

Word Problem 1

4

Aisha painted

together?

3 1 and her sister painted of the wall. What portion of the wall did they paint 5 6

2

3 2 bottle of mango juice and Lisa drank of orange juice. How much juice did 4 9 they consume in all?

3

A rectangular field is

Ram drank

3 2 m long and m wide. Find the perimeter of the rectangular field.

8 5

Multi-step Problems on Adding and Subtracting Fractions

1 Remember, Riya got her first salary? Next month, when she gets her next salary, she spends of her salary 3 1 1 on house rent, of her salary on transportation, of her salary on food.

12 4 Let’s see what fraction of salary is left with her.

1 1 1 4 1 3 4+1+3 8 + + = + + = = 3 12 4 12 12 12 12 12 8 12 8 12 − 8 4 1 Fraction of salary left = 1 − = − = = = 12 12 12 12 12 3 1 Thus, next month Riya saves of her salary.

3 Total fraction of salary spent =

Example 3

3 Mary is baking a cake and needs cup of 4 2 flour. She accidentally adds cups extra, but 5 1 then realizes her mistake and removes 8 cups. How much flour does she have now?

The initial amount of flour Mary needs = 2 cups 5 3 2 15 + 8 23 Total flour = + = = 4 5 20 20 1 Flour removed = cups 8 23 1 46 − 5 41 1 Flour left = − = = =1 20 8 40 40 40 1 Thus, Mary has 1 cups of flour.

40 Extra flour added =

Do It Together

Example 4

3 cups 4

3 Sarah baked a pie and shared with her 8 1 neighbors and with her colleagues. Later, 6 1 she ate of the pie. What fraction of the pie 4 is remaining now?

The whole pie can be considered as a whole.

3 Pie shared with neighbors = ; Pie shared with 8 1 colleagues = 6 1 Pie eaten by Sarah later = 4 3 1 1 9+4+6 Total pie eaten or shared = + + = 8 6 4 24 19 = 24 19 24 − 19 5 Pie left = 1 − = = 24 24 24 5 Thus, of the pie left.

24

3 1 cups to make fruit punch, 6 to make a flavoured jelly and 4 3 drank the remaining. How much did she drink?

Pihu brought 17 cups of juice. She used 4

Total juice with Pihu = 17 cups

3 cups 4 1 Juice used to make flavoured jelly = 6 cups 3 Total juice used to make fruit punch and flavoured jelly = Juice used to make fruit punch = 4

Juice left for drinking =

Do It Yourself 2B 1

1 4 gallons of water. If Nishant drinks gallons of water each day from the jug, then what quantity of 4 7 water will be left in the jug after 3 days?

A jug has 2

Chapter 1 • Operations on Fractions

5

2

1 1 1 km, Muskan jogged 3 km, Neeta jogged 2 km and Avyaan jogged 3 km. What distance did 4 2 6 they all jog together?

3

Shikha studies for 5

4 5

6

Vishwa jogged 3

2 4 hours daily. She devotes 2 hours of her time for science and math. How much time does 3 5 she devote for other subjects?

6 2 An empty box weighing 20 kg. An iron ball weighing 7 kg is placed inside it. What is the total weight of the box?

7 3 2 1 Carlton had 30 yards of string. He uses 11 yards of string to tie a parcel and 6 yards of string to tie a box. How 5 4 much length of string is left with Carlton?

Kashvi recorded the following amount of water used for various activities at home.

Activity

Water used (in litres)

Activity

Water used (in litres)

Bath

20

2 3

Washing car

25

2 5

Washing utensils

10

1 5

Other activities

18

3 4

Laundry

15

1 4

Answer the given questions. a Which activity uses the most water and which activity uses the least water? What is the difference between the amount of water used in these two activities?

b How much total water is used for washing utensils and cars? c What is the total amount of water used? d How much more water is used for bathing than for other activities?

Mean, Median and Multiplication andMode Division of Fractions Real Life Connect

Remember, Riya found out what fraction of her salary she spent and saved?

If she earns ₹60,000 in a month, then let us see how much money she spent and how much she saved.

Multiplying Fractions To find the amount of money spent or saved by Riya, we need to multiply the total money earned by the fraction of the money saved or spent.

Multiplication of a Fraction by a Whole Number The fraction of money spent by Riya = Total salary of Riya = ₹60,000 3000 13 Money spent = × 60, 000 20 1 = ₹39,000

Thus, she has spent ₹39,000.

6

13 of her total salary 20

The fraction of money saved by Riya =

7 of her total salary 20

Total salary of Riya = ₹60,000 7 7 × 60,000 Money saved = × 60,000 = = 21,000 20 20 Thus, Riya saves ₹21,000.

Example 5

Multiply: 4 × =

Do It Together

7 2

Example 6

4 × 7 28 = = 14 2 2

Remember!

Multiply: 30 × 2 = 30 ×

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

4 5

14 420 = = 84 5 5

3 Rajeev went shopping and carried ₹5000 with him. He spent of the total money on shopping.

8 How much money did he spend?

Total amount of money with Rajeev = ₹5000 3 Fraction of money spent = 8 3 Total money spent = ₹5000 × = = ₹ 8 Thus, Rajeev spent __________.

Multiplication of a Fraction by a Fraction Riya keeps half of her saved salary in her account and gives half of her savings to her father for investment.

How much money does she give to her father?

1 7 Fraction of savings given to father = of 2 20 1 7 7 = × = 2 20 40 7 7 Amount given to father = of ₹60,000 = × ₹60,000 = ₹10,500 40 40 Thus, Riya gives ₹10,500 to her father for investment.

Example 7

Jack and Jill had 5 litres of glucose water. Jack consumed

out how much water Jack and Jill consumed individually.

Remember!

Multiplication of numerators N1 N2 N1 × N2 × = D1 D2 D1 × D2 Multiplication of denominators

1 of it and Jill consumed the remaining. Find 5

Total quantity of glucose water = 5 litres

1 Fraction of glucose water consumed by Jack = 5 1 Glucose water consumed by Jack = × 5 = 1 litre 5 Glucose water consumed by Jill = 5 litres – 1 litre = 4 litre Example 8

Multiply a

3 2 × 5 5 Multiply the numerator with the numerator and the denominator with the denominator.

3×2 6 = 5 × 5 25

1 1 × 5 3 1 Convert 4 into improper fraction first, then 5 multiply.

1 21 4 = 5 5

b 4

4 Chapter 1 • Operations on Fractions

1 1 21 1 7 2 × = × = =1 5 3 5 3 5 5

7

Do It Together

Multiply. a 3

1 ×5 2

3

1 = 2 2

3

1 ×5= 2

b 2

2 1 ×5 3 2

2

2 = 3 3

5

1 = 2 2

2

2 1 ×5 = 3 2

Do It Yourself 2C 1

Multiply. a 9×

2

1 3 4 × × 2 4 5

c

2 9 × 3 3

d 1

b 1

4 3 × ×6 9 2

c

1 5 1 × ×3 ×2 4 6 5

d

2 7 1 × 1 − 5 9 2

b

1 1 5 + ×1 4 2 6

c

2

1 of 1000 grams 5

b

1 of an hour 6

c

9 of 51 17

Find. a

5

1 1 × 9 3

Perform the given calculations. a 2

4

b 2

Simplify. a

3

6 5

Answer the given questions.

2 a What is of 18?

9 3 c What is of 2 hours?

4 7 e What is of 5 dozen?

8

3 1 1 5 × × +2 4 2 9 7

5 6 ×2 6 7

1 1 5 ×5× × 6 25 2

d 1

2 5 1 × 2 +3 3 6 3

d 3

1 2 of 7 5

e 5

3 × 21 4

e 2

1 1 1 ×1 ×1 4 3 2

e

5 of a meter 6

16 9 of ?

27 2 2 d How many days are of 7 weeks?

7 2 f How many days are of a leap year?

3 b How much is

Word Problem 1

8

Ameena has 8

did she use?

2 2 litres of oil. She uses only litres of it in making french fries. How much oil 5 3

1 1 litres of milk. He uses only of it and gives the remaining to his friend. Find 5 3 a How much milk did he use? b How much milk was he left with?

2

Ram has 2

3

3 of a pizza. The total weight of the pizza was 8 kg. What 4 weight of pizza did Simran eat? How much pizza is left?

Simran and her friends ate 1

4

Pihu made 30 small cookies. She gave two-thirds of it to her sister and gave a one-fifths of

5

In a library, there are 1500 books. Out of these,

it to her neighbors. How many cookies did she distribute? remaining are English books. Find: a Number of English books

1 3 are science books, are novels and 5 5

b Number of Novels

c Number of science books

Dividing Fractions Remember, Riya gave half of her savings to her father, and the rest she keeps in her bank account. Let’s see what fraction of money she keeps in her bank account.

Division of a Fraction by a Whole Number

7 7 of her salary and keeps half of the savings in her bank account. So, we will divide by 2 to 20 20 find the fraction of the amount kept in the bank account.

7 Let us find ÷2 20 Riya saved

Write the whole number as a fraction.

Write the reciprocal of the fraction.

7 7 2 7 1 7×1 7 ÷2= ÷ = × = = 20 20 1 20 2 20 × 2 40 Multiply the fractions to get the answer.

Reverse the '÷' symbol to '×'.

Let’s see the Keep, Change and Flip method

Example 9

Divide 4 5

7 20

÷

2

Keep

Change

Flip

7 20

×

1 2

4 by 2.

5

Answer

=

Example 10

÷

2

Keep

Change

Flip

4 5

×

1 2

7 40

Divide 2 2

Answer

=

Chapter 1 • Operations on Fractions

4×1 = 4 =2 5 × 2 10 5

2 12 = 5 5

2 by 3.

5

12 12 3 12 1 12 ÷3= ÷ = × = 5 5 1 5 3 15 Reduce

12 4 = 5 5

9

Do It Together

Divide 7

1 by 5 2 7

1 = 2

÷5=

÷

5 = 1

×

Error Alert!

=

Always multiply by the reciprocal while dividing a fraction.

Division of a Whole Number by a Fraction What if we want to divide 2 by half. How many halves do we get?

1 2÷ 2 Write the whole number as a fraction.

2÷

Do It Together

1 2 1 2 2 2×2 4 = ÷ = × = = 2 1 2 1 1 1×1 1 Multiply the fractions to get the answer.

3 Divide 12 by .

4 3 12 3 12 4 12 × 4 48 12 ÷ = ÷ = × = = =16 4 1 4 1 3 1×3 3

2 Divide 9 by 1 .

3 2 1 = 3 9 9 9÷ = ÷ = × 1 1

2 2 1 2 ÷3= × = 3 3 3 9

Write the reciprocal of the fraction.

Reverse the '÷' symbol to '×'.

Example 11

2 6 ÷3= =2 3 3

Example 12

1 Divide 21 by 2 .

3 1 7 2 = 3 3 7 21 7 21 3 21 × 3 63 21 ÷ = ÷ = × = = =9 3 1 3 1 7 1×7 7

=

Division of a Fraction by Another Fraction Now, let us learn how to divide a fraction by a fraction.

1 1 Divide by .

3 9

Example 13

Divide

1 1 ÷ 4 3

1 1 1 3 1×3 3 ÷ = × = = 4 3 4 1 4×1 4

1 3

÷

1 9

Keep

Change

Flip

1 3

×

9 1

Example 14

=

Divide 3 3 3

10

Answer

2 17 = 5 5

1×9 9 = =3 3×1 3

2 1 by 2 5 3 2

1 7 = 3 3

2 1 17 7 17 5 17 × 5 17 3 ÷2 = ÷ = × = = =2 5 3 5 5 5 7 5×7 7 7

Do It Together

Divide 5 5

2 = 3

2 5 by .

3 7 ÷

5 = 7

÷

5 = 7

×

=

Do It Yourself 2D 1

Find the reciprocal of the given numbers. a

2

7 11

b 2

2 ÷ 6 = _____ 3

b

3 5

d 2

c 14 ÷

2 ÷ 66 = _____ 11

2 1 ÷ 5 4

67 4

b 2

5 2 ÷1 8 3

Answer the given questions. a What is 96 divided by the sum of

3

3 40

1 4

e 1

7 8

c 8

1 ÷ 4 = ____ 2

c 1

7 2 by 10 5

Compare using appropriate symbols. (>, <, =) a 3

5

b 18 ÷

Fill in the blanks. a

4

c 3

Divide and simplify. a 15 ÷ 35

3

5 9

d 2÷5

19 4

f

1 6

d 2

10 ÷ 1 = _____ 13

d 1

1 2 by 2 3 27

19 11

e 6÷9

2 3

5 7

12 8 and ?

8 12

4 2 b If product of two numbers is . If one of the numbers is , find the other.

5 3 3 c The perimeter of an equilateral triangle is m. What is the length of each side?

7

Word Problem 1 2

3

The area of a rectangular field is 43

3 1 cm2. If its breadth is 12 cm, find its length.

4 2

1 liters of juice. If the entre juice has to be poured into 3 5 small containers, each of capacity 1 litres, then find the number of containers required.

6 5 Seema bought kg fruits. She packed them equally into 15 packets.

9 a Find the weight of 1 packet of fruit in kg.

A large can of juice contains 18

b She sold 5 packets of fruit. How many kilograms of fruits did she sell?

4

Akshay has a piece of cloth which is 12 cm long. He cuts it into small pieces of length

each. How many pieces of cloth will he have?

Chapter 1 • Operations on Fractions

2 cm 3

11

Multi-step Problems on Operations on Fractions

1 5 of her salary the next month? She gave of the savings to her father for 3 11 investment and kept the remaining savings in her bank account. She earned ₹66,000 that month.

Remember, Riya saved

Let us see what amount of money she keeps in her bank account.

1 × ₹66,000 = ₹22,000 3 5 5 Savings given to father for investment = of the savings = × ₹22,000 = ₹10,000 11 11 Savings deposited in the bank account = Total savings – Savings given to father

Total money saved =

= ₹22,000 − ₹10,000 = ₹12,000

Example 15

Thus, Riya keeps ₹12,000 in her bank account.

3 In a fruit market, of the fruits are mangoes. If the total number of mangoes is 48, the vendor also 5 5 knows that of the remaining fruits are apples. How many apples are there in the market, and what 8 fraction of the total fruits are apples?

3 Fraction of fruits that are mangoes = ; Total number of mangoes = 48 5 5 240 Total number of fruits = × 48 = = 80 3 3 Remaining fruits = 80 – 48 = 32 5 Fraction of remaining fruits that are apples = 8 5 160 Total number of apples = × 32 = = 20 8 8 20 1 Fraction of fruits that are apple = = 80 4

Example 16

4 A man sells orange juice. A medium glass contains 350 mL, while the large is more than the 5 1 2 medium and the small is of large glass. If three people drink 1 small, 1 medium and of a large glass 3 9 respectively, then find the total quantity of juice drunk by three of them.

Quantity of a medium glass of juice = 350 mL 4 4 Quantity of a large glass of juice = 350 mL + of 350 mL = 350 mL + × 350 mL = 350 mL + 280 mL = 630 mL 5 5 1 1 Quantity of a small glass of juice = of large glass of juice = × 630 mL = 210 mL 3 3 2 2 Let us find of a large glass = × 630 mL = 140 mL 9 9 2 Total juice consumed by three people = 1 small glass + 1 medium glass + of large glass 9 = 210 mL + 350 mL + 140 mL = 700 mL

Do It Together

Thus, the total quantity of juice that three people drank is 700 mL. 6 5 Preeti baked a cake and shared with her neighbors and ​ with 16 12 2 her friends. Later, she gave of what was left to her sister. What 5 fraction of the cake is remaining now?

6 Cake shared with neighbors = 16 5 Cake shared with friends = 12 12

Total cake shared = Fraction of cake remaining after sharing = Fraction of cake given to her sister = Fraction of cake remaining after sharing with sister =

Do It Yourself 2E 3 of the people at a restaurant are males. If the number of females is 28 more than males, then how many 7 females are there at the restaurant?

1

1 1 of the total shirts at a shop are printed, of the remaining shirts are striped and rest of the shirts are plain 3 4 shirts. If there are 96 plain shirts then how many total shirts are there?

2

3 2 of the total players at an event are football players. of the football players are female. If there are 72 then how 8 5 many players at the event are non-football players?

3

4

The heights of three lamp-posts L1, L2 and L3 are proportional and taken in the same order. If L1 is 10 m tall and

5

Kevin has 25 m of ribbon. He gives 13 m of ribbon to his sister. He works on a project and uses 1

L3 is 90 m, how tall is L2?

each project. How many projects can he complete?

1 m of ribbon for 2

1 1 2 of money with her on house rent, of the money on food and of the money on other expenses.

5 4 5 She is left with ₹3000. How much money did she have initially?

Rakul spends

6

1 1 gallons per half an hour, while the other leaks at the rate of 4 5 gallons every quarter of an hour. In how many hours will the cistern be empty if it contains 500 gallons of water?

A cistern has two holes. One leaks at the rate of

7

Points to Remember • •

a The numbers of the form , where a and b are whole numbers and b = 0 are called fractions. b To add/subtract unlike fractions, find the LCM of the denominators, then convert them into like fractions and perform addition/subtraction.

•

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

•

When a fraction is used with the word “Of” it acts as an operator. ”Of“ means multiplication.

•

A reciprocal of a fraction can be obtained by interchanging the numerator and denominator. Two fractions are said to be reciprocal of each other if their product is 1.

•

To divide two or more fractions, multiply the first fraction with the reciprocal of other fractions.

Chapter 1 • Operations on Fractions

13

Math Lab Fraction Collage Addition Aim: Adding unlike fractions using hands-on visual representation.

Materials Required: Coloured construction paper, pen, paper, scissors, glue stick Setting: In pair Method:

1 2 3 4

1 Instruct students to draw and label fractions with different colours. Example: in red, 4 1 in blue etc.

3 Students have to carefully cut out the coloured fractional parts.

Students have to combine the cut outs from different colours to create new fractions.

1 1 7 For example, adding (red) and (blue) to get (using colored pencils for clarity).

4 3 12 Instruct students to glue their fraction cutouts representing the addition onto a separate piece of paper, visually representing the addition of unlike fractions.

Chapter Checkup 1

2

Write true or false. a Any fraction divided by 0 is not defined.

______________________

b Associative law is applicable to the division of fractions.

______________________

c The reciprocal of 1 is 1.

______________________

d The commutative law is applicable to the subtraction of fractions.

______________________

Add the given unlike fractions. a

3

7 11 + 9 17

5 2 + 13 39

c

1 5 + 9 6

d

14 25 + 6 4

e

12 64 + 7 14

Find the difference between the given mixed numbers and write the answer in its simplest form. a 11

4

b

1 7 −2 4 6

b 6

4 1 −2 5 7

c 27

10 7 −2 11 9

d 1

6 1 −1 13 54

Fill in the blanks.

5 and its reciprocal is ______.

6 b A fraction is said to be in its lowest form when the ____________ of the numerator and denominator is 1. a The product of

1 is the same as multiplying by ______.

2 d The reciprocal of proper fractions is always greater than ___________. c Dividing by

5

14

Find the reciprocal of the fractions.

1 a 5 b 4

c 101

d

15 17

e

18 5

6

Simplify. a

7

1 3 × 2 5

b 3

2 1 ÷ 9 2

Complete the following.

1 of two hours = ________ minutes 9 1 c 5 of an year = ________ days 5

a

8

Solve. a 1

9

2 4 1 +3 +4 3 5 2

b 23

19 5 − 17 23 11

c 2

13 1 +4 15 5

d 2

15 2 − +1 17 34

2 of a day = ________ hours 3 3 d of a metre = ________ cm 4

b

c 2

6 1 5 × × 7 3 7

d 5

3 2 2 ÷4 ÷1 4 3 3

Answer the given questions.

1 1 1 1 and 2 or quotient of 2 and 4 .

5 4 3 6 4 1 4 b Which is less? The product of 25 and or the sum of 3 and 5 .

75 4 5 2 2 c What is the difference between the quotient of 4 and and the product of and 15?

7 5 1 1 1 d What is the sum of the product of 9 and 1 and the sum of 9 and 7 ?

2 2 3 a Which is more? The product of 3

10 Solve the given questions. a A train covers a distance of 48

3 1 miles in 1 hours. How far does it go in 2 hours?

4 4

2 of x = 26.

13 2 Find the value of x, if 22 × x = 1800 9 3 2 If product of two numbers is . If one of the numbers is , find the other.

5 3 6 If of a number is 180. Find the number.

7 3 47 The product of two numbers is 21 If one number is 7 , find the other.

5 55

b Find the value of x, if c d e f

Word Problems 1

a

2

3

4

3 2 of a cake. She ate of it.

5 3 Find what fraction of cake she ate?

Miya has

b

What fraction of cake is left with her?

a

3 turns about its axis each day. Find 5 How many turns will it complete in a week?

b

How many turns will it complete in one fortnight?

A planet completes 1

4 7 full. When 65 litres of water are drawn from it, it is full. Find how 5 12 much water is left in the tank.

A tank is

1 1 m per s for 5 hours and then at the speed of 3 m 5 4 per s for the next 3 hours. Find the distance travelled by Piyush in 8 hours.

Piyush walks at a speed of 4

Chapter 1 • Operations on Fractions

15

5

1 2 of a distance by train, by bus and walks for the remaining 3 5 distance. If she walks for 16 km, then find the distance travelled by: Himani travels a

6

a bus

b

a train

c

total distance.

A group of students decided to organize a charity event to raise funds for their local animal shelter. They planned to sell custom-made bracelets and donate a fraction of the proceeds to the shelter. Here is how their plan unfolds: They created 3 different types of bracelets: A, B and C.

2 For every 5 bracelets of type A sold, they would donate of the proceeds to the 5 shelter.

3 • For every 3 bracelets of type B sold, they would donate of the proceeds to the 7 shelter.

5 • For every 4 bracelets of type C sold, they would donate of the proceeds to the 8 shelter.

•

During the event: a

They sold 35 bracelets of type A for ₹70 each.

b

They sold 21 bracelets of type B for ₹90 each.

c

They sold 28 bracelets of type C for ₹120 each.

Calculate the total amount donated to the animal shelter from the proceeds of each type of bracelet sale.

16

Operations on Decimals

2

Letʼs Recall Decimals are the numbers which consist of two parts namely, a whole number part and a fractional part separated by a decimal point.

Hundreds

Tens

Ones

2

7

5

PART

• decimal point

WHOLE

Tenths

9

Hundredths Thousandths

1

8

While multiplying a decimal number with 10, 100 or 1000 etc., we move the decimal point as many places to the right as there are zeros in the multiplier.

For example,

On multiplication with 10

On multiplication with 100

23.389 � 10 = 2 3.3 8 9 = 2 3 3.8 9

23.389 � 100 = 2 3.3 8 9 = 2 3 3 8.9

On multiplication with 1000 23.389 � 1000 = 2 3.3 8 9 = 2 3 3 8 9.0 Similarly, while dividing a decimal number with 10, 100 or 1000 etc., we move the decimal point as many places to the left as there are zeros in the divisor.

For example,

On dividing by 10

On dividing by 100

85.3 � 10 = 8 5.3 = 8.5 3

85.3 � 100 = 8 5.3 = 0.8 5 3

On dividing by 1000 85.3 � 100 =

8 5.3 = 0.0 8 5 3

Let's Warm-up Fill in the blanks.

1

351.254 × 100 = _______

4

39.145 ÷ 100 = _______

2

45.123 × 1000 = _______

5

587.2 ÷ 1000 = _______

3

124.12 ÷ 10 = _______ I scored _________ out of 5.

Mean, Median and Mode of Decimals Addition and Subtraction Real Life Connect

Suhani and her mother visited the market to buy fruit and vegetables. The shopkeeper weighed the purchase using a digital scale. They purchased 5.755 kg of vegetables and 7.35 kg of fruits.

Adding and Subtracting Decimals Let us see the total weight of the items purchased by Suhani and her mother.

To find the total weight of items, we need to add.

Here the weights are unlike decimals.

Like decimals are decimals with same number of decimal places after the decimal point. For example, 25.36 and 52.65 are like decimals.

Unlike decimals are decimals with a different number of decimal places after the decimal point. For example, 52.45 and 65.124 are unlike decimals.

Error Alert!

The misalignment of decimal points leads to an incorrect answer.

23.53 + 1.0 39.53

Let us see how to add decimals that are unlike.

Step 1: Convert the decimals into like decimals.

Here, 5.755 and 7.35 as 7.350

Step 3: Add as whole numbers are added.

T O .

t h th

5 + 7

7 5 5 3 5 0

1

1 3

. .

23.53 + 1.0 25.13

Step 2: Write the digits in the column method aligning the decimal point and the digits.

T O .

5 . + 7 .

t h th 7 5 5 3 5 0

Step 4: In the sum, put the decimal point directly below the other decimal points.

1

T O .

t h th

1 0 5

5 + 7

7 5 5 3 5 0

1

1 3

. .

.

1

1 0 5

We can use the same method to subtract decimals.

Let us now find the difference in the weight of the fruit and vegetables purchased.

Step 1: Convert the decimals into like decimals.

Here, 5.755 and 7.35 as 7.350

Step 2: Write the digits in the column method aligning the decimal point and the digits.

T O .

7 .

– 5 .

18

t h th 3 5 0 7 5 5

Step 3: Subtract as whole numbers are subtracted.

O .

6

7 – 5 1 Example 1

. .

t h th 2

O .

1

3 5 0 7 5 5 5 9 5

6

7 – 5

1

1 2 1 + 3

2

2 1 4 2

1

5 3 8 6

8 1 3

. . . .

.

Example 2

2 5 5 1

3 2 8 2

2

4

1

3 5 0 7 5 5

.

5 9 5

Subtract 235.28 from 365

t h th 1

t h th

. .

1

What is the sum of 125.23, 213.523, 148.58 and 326.125?

H T O .

Example 3

4

Step 4: In the difference, put the decimal point directly below the other decimal points.

H T O .

5

0 3 0 5

5

4

3 6 5 – 2 3 5 1 2 9

. . .

t h

9 10

0 0 2 8 7 2

4 5 8

The length of a rope is 102 m and 80 cm. If a piece measuring 59.35 meters is cut from it, what is the length of the remaining rope?

Length of rope = 102 m 80 cm As the measure is not in decimal, we can write it in decimal by converting to the higher unit as = 102.8 m Length of rope cut down = 59.35 m Remaining rope = 102.80 – 59.35 = 43.45 m

Do It Together

Add the sum of 367.25, 125.2 and 512 to the difference of 563.24 and 321.

Sum = 367.25 + 125.2 + 512 = _____ .... (1) Difference = 563.24 − 321 = ______ ...... (2)

H 3 1 + 5

T 6 2 1

O 7 5 2

. . . .

t 2 2 0

h 5 0 0

H T O .

5 6 3 .

– 3 2 1 .

t h 2 4 0 0

Adding (1) and (2) gives ____ + _____ = _____

Do It Yourself 3A 1

2

Add. a 223.25 and 12.9

b 315.36 and 218.234

c 128.2 and 236.54

d 332, 127.456 and 122.26

e 136.23, 556.14 and 25.125

f 152.214, 235.3 and 365.28

a 322.36 from 438.2

b 132.89 from 325.566

c 118.5 from 732.55

d 438.236 from 752.23

e 856.18 from 998.856

f 1065.235 from 1189.15

Subtract.

Chapter 2 • Operations on Decimals

19

3

Fill in the blanks. a 236.45 + ____ = 568.2

b 514.7 − _____ = 241.23

c ___ + 321 = 512.47

d ____ − 475 = 251.487

e 785.5 + 324.21 = _____

f 987 – 562.26 = _____

4

What should be added to 389.265 to get 625.56?

5

By how much should 235.125 be decreased to get 126.24?

6

Simplify 126.256 + 325.46 – 195.264 + 326.152 – 256.369

7

Subtract the difference of 469.251 and 829.23 from the sum of 356.21 and 585.236.

8

A recipe requires 3.75 cups of flour, 2.5 cups of sugar, and 1.25 cups of butter. How much of these ingredients are

9

The distance travelled (in km) by Mohit in a week was recorded on a bar graph.

86.00

84.00

84.5

82.75

82.00

81.5

80.5

80.00

79.75

78.25

78.00

76.55

76.00 74.00

Su

nd

ay

y da tur Sa

da Fri

sd ur Th

y

ay

da es dn We

da es Tu

nd Mo

y

y

72.00 ay

Distance Travelled in km

needed in total for the recipe?

Day

a What is the total distance travelled by Mohit in the week? b What is the difference between the distance travelled on the last 3 days of the week

Word Problem 1

Ravi goes to buy a big box of sweets. He purchases 10.5 kg of sweets and the weight of the

2

Vineet purchased 2.5 kg of coffee from the market. He uses 1 kg 75 g of it. What weight of

3

A delivery truck had 56.75 gallons of fuel in the morning. After a long trip, it had 32.5

4

Swati bought 7 kg 450 g of sugar,2.35 kg of rice

empty box is 1 kg 350 g. What is the total weight of the sweets and the box? coffee does he have left?

gallons left. How much fuel did the truck use during the journey?

altogether?

5

The average temperature in Amsterdam for 5 months is given in the line graph. What is the difference between the temperature of the hottest and coldest month?

20

Temperature In ℃

and 5 kg 700 g of flour. How much did she buy

25

17.6

20 15 10 5 0

9.7

20.2

12.8

5.5

FEB

March

April Month

May

June

Multi-step Word Problems on Adding and Subtracting Decimals Suhani and her mother spent ₹596.75 on purchasing vegetables, and ₹895.45 on purchasing fruit. They also made miscellaneous purchases of ₹1236. If they gave ₹3,000 to the shopkeeper, how much change would they receive?

Th H T O .

2

1

5 8 1 + 1 2 2 7

1

9 9 3 2

6 5 6 8

t h 1

. . . .

7 4 0 2

5 5 0 0

Th H T O .

2

9

9

9

3 0 0 0 – 2 7 2 8 2 7 1

. .

.

t 10

0 2

8

Amount spent on vegetables = ₹596.75 Amount spent on fruit = ₹895.45

Amount spent on miscellaneous purchases = ₹1236

Total amount spent = 596.75 + 895.45 + 1236 = ₹2728.2

Change received from the shopkeeper = 3000 – 2728.2 = ₹271.80 Example 4

A drum has 59.55 kg of rice in it. A pack of 25.5 kg of rice is emptied into the drum. After this, 312 kg of 5 rice is used. What is the weight of the rice left in the drum?

T O .

1

1

5 9 + 2 5 8 5

. .

.

t h 5 5 5

0 5

T O .

4

8 5 – 3 1 5 3

. .

.

t h 10

0 5 4 0

6 5

Quantity of rice in the drum = 59.55 kg; Quantity of rice in the packet = 25.5 kg Total quantity of rice after emptying the packet = 59.55 + 25.5 = 85.05 kg Quantity of rice used = 312 = 157 = 31.4 kg 5 5 Quantity of rice left in the drum = 85.05 – 31.4 = 53.65 kg Example 5

Mayra dedicated 7 hours to finishing her maths homework, 1.2 hours to complete her science homework, 4 7 and spent hours less time on Social Studies homework compared to the time she spent on the maths 10 homework. What was the total amount of time she spent completing her homework that day?

Time to finish Maths homework 7 = 1.75 hours 4 Time to finish science homework = 1.2 hours

Time to finish social studies homework = 7 = 0.7 hours less than maths homework = 1.75 – 0.7 = 1.05 hours 10 Total time to finish the homework = 1.75 + 1.2 + 1.05 = 4 hours Do It Together

Sushant had 182 cups of sugar. He used 2.25 cups to make milk shake, 63 for making a dessert, 1.45 5 4 cups for baking a cake. How much sugar is left?

Total cups of sugar = 182 = 18.4 cups 5 3 Sugar used = 2.25 + 6 + 1.45 = 2.25 + _________ + 1.45 = _________ cups.

4 Sugar left = 18.4 − _________ = _________ cups.

Chapter 2 • Operations on Decimals

21

Do It Yourself 2B 1

A palm tree is 42.68 feet tall while a mango tree is 15.23 feet shorter than the palm tree. What is the total height

2

Ronita pays ₹635.80 every month for cable and internet. She came to know about a new plan by another company

of both trees?

that offered ₹452.75 for internet plus an additional ₹126.55 for cable. If Ronita switches to the plan, how much will she save every month?

3

Sumit downloaded three applications on his mobile that were 986.48 kb in total. If one of the apps was 256.36 kb

4

Rhea donated 35.65 kg of sugar, 78 kg 540 g flour, 17 kg 320 g rice and some pulses to an NGO. If the total weight

5

Sam saved ₹3652.45 in June and ₹4578.58 in July. He then purchased a bicycle worth ₹2468.75 and a helmet

6

There are 2 litres 125 mL of juice in jug A while there is 1 litre 350 mL juice in jug B. If both the jugs together can

7

The amount deposited and withdrawn by Manish in the initial four months of the year are shown below. What is

and the other was 485.84 kb, how big was the third application?

of the items donated was 142.63 kg, what was the weight of the pulses donated?

wjaborth ₹689.48. How much money was he left with at the end of July?

hold 4.75 litres of juice, how much more juice can be poured into both the jugs?

the difference in the amount deposited and withdrawn in the four months?

Amount in ₹

12000 10000 8000 6000

10512.25

5423.58

4000

9684.5

10365

6842.75

3754.36

4365.45

February

March

April

8523

2000 0

January

Month Money deposited Money Withdrawn

22

Word Problem 1

Rita went from place X to place Y and then to place Z. X is 14.05 km from Y and Y is 12.7 km

from Z. Aman went from place X to place A and then to place Z. A is 29.3 km from X and Z is 21.45 km from A. Who travelled farther and by how much?

2

Sam ordered a coffee and a waffle. Find the change he will receive if he paid $10.

Coffee: $1.46

Garlic bread: $3.95 Ice tea: $2.89 Waffle: $0.85

Multiplication and Division of Decimals Real Life Connect

Naman and his father are patiently waiting their turn at the petrol pump. Naman notices the displayed prices of various fuels on the board.

Naman: Dad, how much fuel are we putting in the car today?

Father: We're going to fill our car with 35.5 litres of diesel.

Naman's curiosity is sparked as he begins to calculate the amount they will need to pay at the pump.

Petrol –101.94 Diesel – 87.89 LPG –

77.85

CNG –

81.5

Multiplying and Dividing Decimals Let us find the amount paid by Naman’s father.

To find the amount paid by Naman’s father, we need to multiply. To multiply two decimal numbers we follow the steps.

Chapter 2 • Operations on Decimals

23

Step 1: Multiply the decimal numbers ignoring the decimal points, just like whole numbers.

4 4 3 + 2 6 3 3 1 2

Step 2: Count the total number of decimal places and place the decimal point in the product accordingly.

Hence, 87.89 × 35.5 = ₹3120.095

8 � 3 9 6 0

7 3 9 4 7 0

8 5 4 5 0 9

9 5 5 0 0 5

Remember!

The product should have the same number of decimals place as the sum of the number of decimal places in both the multiplicand and multiplier.

Naman saw that one of the rickshaw drivers paid ₹1018.75 for filling with CNG. He started wondering, how much CNG this rickshaw driver must have put into his auto rickshaw tank.

Let us find out!

Step 1: Make the divisor a natural number by multiplying the divisor and the dividend by a power of 10.

81.5 × 10

815

1018.75 × 10

10187.5

Step 2: Place the decimal point directly above the decimal point in the dividend.

Step 3: Divide the whole number part and then the tenth part.

Example 6

Multiply. a 457.2 and 132

+

4 � 9 1 3 7 4 5 7 6 0 3

5 1 1 1 2 5

7 3 4 6 0 0

2 2 4 0 0 4

457.2 × 132 = 60350.4 Example 7

b 102.925 and 4.28 1 0 2 � 8 2 3 2 0 5 8 + 4 1 1 7 0 4 4 0 5 1

9 4 4 5 0 9

2 2 0 0 0 0

5 8 0 0 0 0

102.925 × 4.28 = 440.51900

Divide. a 3259.36 by 26 125.36 26 3259.36 – 26 65 – 52 139 – 130 93 – 78 156 – 156 0

Do It Together

b 0.888 by 1.11 0.8 111 88.8 – 00 888 – 888 0

1.11 × 100 = 111; 0.888 × 100 = 88.8

Think and Tell Do we get a natural number on dividing 2 decimal numbers?

Find the missing dimension of the given rectangle with area 143.75 cm2.

Area of rectangle = 143.75 cm2; Breadth of rectangle = 11.5 cm Area = _________ = _________ Breadth Hence, the length of the rectangle is _________ cm.

Length =

24

12.5 815 10187.5 – 815 2037 – 1630 4075 – 4075 0

11.5 cm ?

Do It Yourself 8C 1

2

Find the product. a 0.12 × 15

b 112.2 × 6.5

c 32.1 × 1.33

d 5.014 × 25

e 52.25 × 5.23

f 122.56 × 5.21

g 322.15 × 2.24

h 21.16 × 12.83

If 365 × 124 = 45260, then find the following without actual multiplication. a 36.5 × 124

3

b 3.65 × 12.4

c 36.5 × 1.24

d 3.65 × 1.24

a 0.165 ÷ 1.5

b 42.7 ÷ 14

c 49.08 ÷ 0.02

d 88.8 ÷ 2.22

e 371.68 ÷ 16

f 164.84 ÷ 13

g 23.85 ÷ 1.8

h 487.36 ÷ 3.2

k 2039.96 ÷ 5.2

l

Find the quotient.

i 4

1117.5 ÷ 12.5

j

115.542 ÷ 4.9

4756.05 ÷ 58.5

Fill in the blanks. a 68.527 × _____ = 23.63

b _____ × 1.6 = 52

c 2177.35 ÷ _____ = 62.21

d 74.2 × 12.3 = _____

e 984.576 ÷ _____ = 25.6

f 616.54 ÷ 14.5 = _____

5

The product of two decimals is 3538.29. If one of them is 41.5, find the other.

6

Find the area and perimeter of the square.

7

3 Find the cost of 45 litres of milk at the rate of ₹54.5 per litre.

4

8

2.35 m of cloth is needed to make a shirt. How many metres of cloth are needed to make 48 such shirts?

9

A ribbon 30 m 80 cm long is to be divided into 11 equal pieces. Find the length of each piece.

124.75 cm

10 A car moving at a constant speed covers a distance of 452.8 km in 8 hours. Find the distance covered by the car in 4 2 hours.

5

Word Problem 1

Ravi went to the grocery store and bought 12 kg 450 g of rice at a cost of ₹52.75 per

2

If 2m 25 cm of cloth is required to stitch a pair of pants, how many pairs of pants can be

3

A grocery owner bought some bags of rice each weighing 97.5 kg. If the total weight of all

4

Mohan has a rectangular plot of area 875.25 square metres. He wishes to divide it equally

kilogram. What is the total cost of the rice bought by Ravi? stitched with a piece of cloth that is 20.25 m long?

the bags is 682.5 kg, find the number of bags of rice bought by the grocery owner. among his 1 son and 3 daughters. How much of the plot will each receive?

Chapter 2 • Operations on Decimals

25

Multi-step Word Problems on Multiplying and Dividing Decimals Naman’s father purchased a computer system for ₹60,225 with an additional tax of ₹10,840.5. He made a payment of ₹12,000 and paid the rest of the amount in six equal installments. What is the amount paid by Naman’s father in each installment.

Total cost of the computer system = 60,225 + 10,840.5 = ₹71,065.5 Balance amount after paying ₹12000 = ₹71,065.5 − ₹12,000 = ₹59,065.5 59065.5 Amount paid in each installment = = ₹9844.25 6 Hence Naman’s father paid ₹9844.25 in each installment.

Example 8

A fruit vendor is selling oranges at ₹35.50 per kg and apples at ₹86 per kg. If Meera buys 21 kg oranges and 1.75 kg apples, what will be the total cost she4needs to pay?

Did You Know?

The decimal system of currency in India was introduced on 1 April 1957.

21 kg oranges = 2kg + 0.25 kg = 2.25 kg 4 Cost of 2.25 kg oranges at ₹35.50 per kg = 2.25 × 35.5 = ₹79.875 Cost of 1.75 kg apples at ₹86 per kg = 1.75 × 86 = ₹150.5 Total amount paid = 79.875 + 150.5 = ₹230.375 Example 9

Fifteen individuals visited a restaurant. Each one of them ordered the items shown in the table. While paying the total bill, they found that five members of the group forgot to bring money. To cover the bill, how much additional money did the remaining individuals need to contribute?

Number of individuals = 15; Cost of meal per person = 25.45 + 34.95 + 70.25 + 15 = ₹145.65 Total bill = 15 × 145.65= ₹2184.75 As 5 people forgot to bring the money, the amount would be paid by 10 people Amount paid by each individual = 2184.75 ÷ 10 = ₹218.475

Item

Price

Idly

₹25.45

Vada

₹34.95

Dosa

₹70.25

Tea

₹15

Additional amount paid by remaining individual = 218.475 – 145.65 = ₹72.825 Do It Together

During a visit to a history museum, a group comprised of 12 adults and 5 children purchased tickets.

With each adult ticket priced at ₹125.5 and each child's ticket priced at ₹75.5, what is the overall difference in the total cost of the tickets for the adults and children combined?

Price of 12 adult tickets at ₹125.5 per ticket = 12 × 125.5 =_____ Price of 5 child ticket at ₹75.5 per ticket = 5 × 75.5 = _____ Difference in the cost = _____ − _____ = ₹_______

26

Do It Yourself 8D 1

Suhani bought 7 kg 650 g sugar at ₹38.5 per kg, 2 kg 485 g rice at ₹62.75 per kg and 5 kg 725 g of flour at

2

Kim bought 25.5 litres of milk. He fills 8 bottles equally with the milk, with 1 1/2 litre of milk left. Find the volume

3

Kunal has a rectangular garden measuring 5.8 metres by 7.25 m. What is the cost of fencing the garden if the cost

4

Mayra bought 2 pairs of shoes for ₹525.75 a pair, three t-shirts for ₹263.50 each, five pair of socks for ₹50.25 a

5

Sunish bought a music system for ₹25,526.75. He made a down payment of ₹10,250 and paid the rest amount in

6

Sam used some cloth to make 8 banners and 3 tablecloths. He used 2 m 75 cm of cloth for each banner and

₹45 per kg. What is the total amount paid by Suhani?

of milk in each bottle.

of fencing is ₹58.75 per metre?

pair and one bag for ₹755. What was the total cost of the items purchased?

15 equal installments. What is the amount paid in each installment?

1.55 m of cloth for each tablecloth. Find the total length of cloth used. Also find the cost of the cloth used if the cost of one m cloth is ₹75.5.

7

Sharvil purchased 25 notebooks at ₹45.65 each, 3 boxes of pen at ₹520.75 each and 2 boxes of markers at

8

Suman bought 7 kg of tomatoes for ₹250.25. How much will Seema pay for 4 and a half kg of tomatoes?

9

Saumya is making a report on the computer. She wants to place a figure which is 12.75 cm wide so that it appears

₹368.15 each. If the shopkeeper gave ₹560.2 as change, how much amount was given to the shopkeeper by Sharvil?

centered on the page. The width of the page is 22.63 cm. How much margin should she leave at both the ends so that the figure is centred?

Word Problem 1

Krishvi bought 3 pair of gloves for ₹234.75 and two pairs of socks for ₹165. How many pairs of gloves and socks can be bought for ₹1286?

Points to Remember •

Decimals can be added or subtracted by converting them to like decimals and aligning the decimal points in the column method.

•

To multiply decimal numbers Multiply as whole numbers are multiplied Count the total number of decimal points Place the same number of decimal points in the product.

•

To divide decimal numbers Convert the divisor to a whole number just above the dividend Divide as whole numbers are divided

Chapter 2 • Operations on Decimals

Place the decimal point

27

Math Lab Setting: In groups of 5 Materials Required: Decimal domino cards (create or print cards with a decimal equation on one card and its solution on another card), Timer Method:

1

Distribute an equal number of decimal domino cards (both equation and solution) to each group.

2

Start the timer and ask the groups to match the equations with their corresponding solutions by placing the cards next to each other in a domino-like chain.

3

The team with the highest number of correct solution matches in the least time will win the game!

Chapter Checkup 1

2

3

4

28

Add. a 122.56 + 36.2

b 321.4 + 12.63

c 16.523 + 132.29

d 225.213 + 88.9

e 536.3 + 311.236

f 524.8 + 333.69

g 525.236 + 226.21

h 756.59 + 413.5 + 623.12

a 132.56 − 45.2

b 225.3 – 152.35

c 267.523 – 159.47

d 347.14 − 178.5

e 336.8 – 285.123

f 586.4 – 387.25

g 658.412 – 247.48

h 745.14−398.475

a 212.5 and 35

b 110.52 and 21.5

c 156.52 and 12.29

d 315.13 and 81.2

e 136.3 and 125.2

f 624.82 and 32.55

g 312.1 and 15.24

h 362.12 and 213.5

a 304.8 by 12

b 23.94 by 6.3

c 90.15 by 1.5

d 498.8 by 4

e 100.32 by 80

f 42.7 by 14

g 3.024 by 0.36

h 564.975 by 15.5

Subtract.

Multiply.

Divide.

5

Aarav scored 452.65 marks out of 600 in the final examination. How many marks did he lose?

6

There is 0.625 kg of powdered milk in each tin. If a carton contains 12 tins, find the total mass of powdered milk in

7

Sack A contains 125.35 kg of rice, sack B contains 25.65 kg more rice than sack A and sack C contains 13.49 kg less

8

The daily consumption of milk in a house is 3.25 litres. How much milk will be consumed in a year?

9

A rectangular garden measures 13.52 metres in width and 17.36 metres in length. What is the area and perimeter

the carton.

rice than sack B. What is the total weight of the rice in all three sacks?

of the garden?

10 Considering an average growth rate of 1 cm 8 mm per month, what would be the total length of a strand of hair after three years of growth?

1 11 Kavya earns ₹523.65 per hour for the first 40 hours she works in a week. She earns 1 times that amount per 2 1 hour for each hour beyond 40 hours in a week. How much will she earn if she worked for 52 hours?

2 12 Abhinav went to a restaurant with his family. The family ordered 3 veg bowls, 2 apple pies, and 4 milk shakes. a How much did the family spend at the restaurant? b How much change will they receive if paid an amount of ₹1000?

Item

Rate

Veg bowl

₹155.26

Apple pie

₹62.37

Milk shake

₹89.49

Word Problems 1

The watermelon bought by Rani is 3 times as heavy as the papaya bought by Manya.

If the watermelon bought by Rani weighs 4.2 kg, what is the weight of the papaya?

2

The weight of a baby elephant was 118.99 kg. After two years, his weight increased by 109.85 kg. Find the weight of elephant after two years.

3

Ravi bought 8.6 kg of sugar. He poured the sugar equally into 5 bottles. There was 0.35 kg of sugar left over. What was the mass of sugar in 1 bottle?

4

The graph shows the amount saved by Monika during 5 weeks.

1800

1563.75

Amount Saved In ₹

1600 1400 1200 1000

1254

1365.78 1026.65

1058.25

800 600 400 200 0

Week 1

Week 2

Week 3

Week 4

Week 5

Week

a What is the total amount saved during the five weeks?

b What is the difference between the lowers and the highest amount saved?

5

Ruchi bought 1 dozen cupcakes for ₹519.60. Her mother asked her to buy 5 more

6

Mohit's family went on a road trip. They got to point A on the first day and point B on the

cupcakes. How much did Ruchi pay for the 5 cupcakes?

second day. How many more kilometres do they need to drive to reach their destination?

If their car travels 11.5 km per litre, how many litres of fuel will they need for the trip?

Car

A 150.25 km

B 175.50 km

?

450.75 km

Chapter 2 • Operations on Decimals

29

3 Integers Letʼs Recall Integers are a collection of whole numbers and negative numbers. Integers does not include fractional part, similar to whole numbers .

Negative integers

–9

–8

–7

–6

–5

–4

–3

Positive integers

–2

–1

0

+1

+2

+3

+4

+5

+6

+7

+8

+9

Zero (Origin)

For example, temperatures above 0° are considered as positive temperatures and temperature below 0° are considered as negative temperatures.

We can compare integers using certain rules as given below: Rule 1:

Rule 2:

Positive integer > Negative integer

Rule 3:

Positive integer with greater

Example: 25 > – 45

Negative integer with smaller

numerical value > Positive integer

numerical value > Negative integer

Example: 45 > 26

Example: –25 > –52

with smaller numerical value

with greater numerical value

We can also arrange integers in ascending or descending order using the above comparison rules as: Smaller to greater

–9

–8

–7

–6

–5

–4

–3

–2

–1

0

+1

+2

+3

+4

+5

+6

+7

+8

+9

Greater to smaller

For example, Ascending order: – 5 < –2 < 1 < 5 < 9 Descending order: 8 > 5 > 2 > –4 > –10

Let’s Warm-up Fill in the blanks with > or < sign.

1

25 _____ 89

2

–32 _____ –16

4

15 _____ 8 _____ 0 _____ –12 _____ –36

5

–14 _____ –9 _____ –1 _____ 13 _____ 20

3

–12 _____ 5

I scored _________ out of 5.

Addition and Subtraction of Integers Real Life Connect

Sara writes down the money she receives and spends every week in a notebook.

Sara: Mom, I’ve written down the money you and dad gave me this week. I’ve also written down the amount I have spent from it.

Money Received

Money Spent

From mom : `90

On notebooks : `60

From dad : `70

On crayons : `50

Mom: Great Sara! How much have you saved this week?

Sara quickly calculates her savings and tells it to her mom.

Addition of Integers We have studied the addition of integers with like and unlike signs in our previous class. Let us recall it.

Rule 1: For addition of integers with like sign, Step 1 Add the absolute values of the integers.

Step 2 Place the common sign before the sum.

Remember!

The absolute value (shown by symbol ||) of an integer is the numerical value of the integer regardless of the sign.

In the case of Sara, the total money received and spent can be given as: |+90| + |+70| = 90 + 70 = 160

Think and Tell

|–60|+|–50|= 60 + 50 = 110

Why have we not put a

As both the integers have (–) sign,

(+) sign before 160?

∴(–60)+(–50) = (–110)

Rule 2: For addition of integers with unlike signs, Step 1 Subtract the smaller absolute value from the greater absolute value.

Step 2

Remember!

Money received is shown by a (+) sign.

Money spent is shown by a (-) sign.

Place the sign of the integer with the greater absolute value before the difference.

The money left could be given as (160) + (–110) or, |+160| – |–110| = 160 – 110 = 50 (as the integer with the greater absolute value has a positive sign)

Chapter 3 • Integers

31

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.

Commutative Property For any two integers a and b, a + b = b + a

Associative Property For any three integers a, b, and c, (a + b) + c = a + (b + c)

Existence of additive identity For integer a, a + 0 = 0 + a = a

Existence of additive inverse For integer a, another integer –a, exists, so that: a + (–a) = 0

Example

Rohan opened a bank account by depositing `1,500 in his account in June 2021.

He deposited `750 in July 2021 and withdrew `1,100 in August 2021. Find his balance.

Amount deposited in June = `1,500 Amount deposited in July = `750 Total amount deposited in June and July = `1,500 + `750 = `2,250 Amount withdrawn = `1,100 = –1100 (as amount withdrawn is denoted by negative sign) Balance = 2,250 + (–1100) = 2,250 – 1,100 = 1,150 The balance is ₹1,150.

Do It Together

The temperature of water in a bowl is 85 ℃. It dropped by 33 ℃ after 20 minutes. The temperature dropped by a further 13 ℃ in the next 10 minutes. What is the temperature of the water after 30 minutes?

Temperature of water in the bowl = 85 ℃ Temperature drop in 20 minutes = _________ Temperature drop in the next 10 minutes = –13 ℃ Total temperature drop = (–33) + (–13) = _________ Temperature after 30 minutes = 85 ℃ + _________ = _________

32

Did You Know?

Indian Mathematicians, Aryabhata and Brahmagupta, made significant contributions to the understanding and calculation of integers.

Subtraction of Integers We have recalled the addition of integers. Let us now recall the subtraction of two integers.

If a and b are two integers, then a – b = a + (–b), i.e. to subtract an integer b from another integer a, add the additive inverse of b to a and keep the sign of the integer with the greater absolute value.

Let us subtract (–12) from (–15) Here, a = (–15); b = (–12) (–15) – (–12) = (–15) + (additive inverse of (–12)) = (–15) + (12) = (–3)

Remember!

Subtracting a negative number is equivalent to adding a positive number.

Properties of Subtraction of Integers Closure Property If a and b are two integers, then a – b = c, where c will always be an integer.

Commutative Property For any two integers a and b, a – b ≠ b – a

Associative Property For any three integers a, b, and c, (a – b) – c ≠ a – (b – c)

Subtraction Property of Zero It states that subtracting zero from any integer leaves the integer unchanged.

In general, a – 0 = a (where a is an integer).

Example

The record high temperature of Canada is 2 ℃ and the record low is –18℃. What is the difference in the high and the low temperature?

Record high = 2 ℃ Record low = –18 ℃.

Difference in temperature = 2 – (–18) = 2 + (18) = 20 ℃ The difference in high and low temperature is 20 ℃.

Do It Together

Mt. Everest, the highest peak in Asia, is 29,029 feet above sea level. The Assal lake in Africa is 510 feet below sea level. What is the difference in level between these two places?

Height of Mt. Everest above sea level: 29,029 feet Depth of Assal lake below sea level = ______ Difference in depth = (29,028) – (–510) = ______

Chapter 3 • Integers

33

Do It Yourself 1A 1

2

3

4

5

Add. a 5 and –9

b 15 and 14

c –28 and 59

d (–23) and 12

e 32 and –122

f –136 and –25

g –152 and 365

h 158 and –125

a –7 + 3

b 12 + (–5)

c –19 + (–14)

d 20 + (–30)

e –69 + 81

f (–123) + (–235)

g 565 + (–125)

h (–485) + (–325)

a 8 from –9

b –12 from 17

c 25 from –32

d 38 from –20

e –36 from –98

f 54 from 223

g 136 from –182

h –214 from 156

Solve the given problems.

Subtract.

Determine the missing numbers in the equations. a 12 + _____ = 18

b _____ – (–17) = 10

c 956 – _____ = 422

d 215 – 136 = _____

e –548 + _____ = –267

f _____ – 192 = 564

g 705 + _____ = 1202

h 815 – (–125) = _____

Fill in the blanks. a The sum of –5 and 8 is ________. b Subtracting 15 from –20 gives ________. c The absolute value of –10 is ________. d If I owe `50 and I receive `30, my new balance is ________. e If I have a gain of `310 and a loss of `125, my net result is ________. f Adding the additive inverse of –8 to –8 gives ________.

6

Subtract 2369 from the sum of –3652 and 5864.

7

State True or False.

8

a The commutative property of addition holds true for integers.

____________________

b The associative property of subtraction holds true for integers.

____________________

c The commutative property of subtraction holds true for integers.

____________________

d The associative property of addition holds true for integers.

____________________

e The distributive property holds true for the addition of integers.

____________________

Fill in the blanks using the properties. Also name the property used. a (21) + (13) = _________ + (21) ; Property used - ____________________ b _________ + 0 = –3 ; Property used - ____________________ c (–35 + 13) + _________ = –35 + (_________+ (–16)) ; Property used - ____________________ d (–23) – _________ = –23 ; Property used - ____________________

34

Word Problems 1

A store had 754 items in stock. They received a shipment of 325 more items. How many items are there now in total?

2

The altitude of a mountain peak is 2,500

metres. A hiker climbs to an altitude of 1,200 metres and then descends to an altitude of 500 metres. What is the total change in altitude experienced by the hiker?

3

A basketball team scores 80 points in the

first quarter, loses 25 points in the second quarter, and then scores 15 points in the

third quarter. In the final quarter, they score 10 more points than they have lost. What is the team’s total score?

Multiplication and Division of Integers Real Life Connect

Rahul and Megha are watching a submarine show on T.V. They listen to an announcement that the submarine has descended 14 km in 20 minutes.

Rahul: Wow! The speed of submarine is really fast. I wish I could travel in such a submarine some day.

Megha: That’s true, Rahul! Although I’m not really fond of underwater adventures, I’m wondering how deep this submarine will travel in the next 40 minutes.

Multiplication of Integers The submarine descends 14 kilometers in 20 minutes. The sea level is 0, then the distance above the sea is in a positive direction. Similarly, the distance below the surface can be taken as negative.

We can say that the submarine travels –14 kilometres in 20 minutes.

Therefore it travels –28 kilometres in 40 minutes as shown.

Starting from 0, we always move to the negative side (in this case, the downward direction) when multiplying integers of unlike signs.

Chapter 3 • Integers

0

Multiplying integers with unlike signs

(sea level)

20 minutes

-14 kms (distance below sea level) 40 minutes -28 kms (distance below sea surface)

35

Multiplying integers with like signs Case 1: When both the integers are positive: Assume that there are two people on the same side of the door. The push represents the positive sign and the pull represents the negative sign.

When both people push the door, the door moves in the pushed direction! This signifies that the product of two positive numbers is always positive!

Case 2: When both the integers are negative: As we know, the negative sign signifies the pull direction. It is the opposite direction to the push direction.

Now, when both people try to pull the door, they will still end up moving the door in the same direction and ʺhelpʺ the pull movement.

This signifies that the product of two negative numbers is always positive!

The above discussion leads us to the following two rules of multiplication of integers.

Rule 1: The product of two integers with unlike signs is always negative.

Rule 2: The product of two integers with like signs is always positive.

Multiplication of more than two negative integers Let us now multiply three integers, say –5, –12 and –7, and see the result.

Here, we first multiply the initial two numbers, i.e., –5 and –12.

(–5) × (–12) = 60 (The product will be positive as two integers with like signs are multiplied.)

Now multiply the result with the third number, i.e., 60 × (–7).

60 × (–7) = –420 (The product will be negative as two integers with unlike signs are multiplied.)

The final result has a negative sign when we multiply 3 negative integers.

Let us now multiply the above result (– 420) with another negative integer (–2).

(–420) × (–2) = 840 Here, the final result is positive when we multiply 4 negative integers.

The above discussion leads us to the conclusion that: If the number of negative integers multiplied are even, then the product is positive.

If the number of negative integers multiplied are odd, then the product is negative.

Example

What will you get on multiplying –5, –6, –2 and –4?

We will first multiply the initial two numbers i.e., –5 and –6.

(–5) × (–6) = 30 (as two integers with like sign are multiplied) Multiply the result with the third number, i.e., 30 × –2.

36

Think and Tell What will be the sign of the final product when 99 negative integers are multiplied?

30 × (–2) = –60 (as two integers with unlike signs are multiplied) Now multiply the result with the fourth number i.e. (–60) × (–4).

(–60) × (–4) = 240 (as two integers with like signs are multiplied) Do It Together

Error Alert!

Don’t forget to place the correct sign (+ or –) before the resultant product.

A store loses ₹125 a day for six days. How much money has been lost in total over these six days?

Money lost by the store every day = _____ Money lost in 6 days = (–125) × _____ = _____

Properties of Multiplication of Integers Closure Property

Commutative Property

If a and b are two integers than

For any two integers a and b,

Associative Property For any three integers a, b, and c,

a×b=c

a × b = b × a.

(a × b) × c = a × (b × c)

where c will always be an integer.

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

This means that when multiplying three or more integers, the order in which the multiplications are performed does not change the final product.

For example, a 13 × (–3) = -39 b (–25) × (–9) = +225

For example,

In both the above cases, the result (15) × (–3) = (–3) × (15) = –45 of the multiplication is an integer.

Distributive property of multiplication of integers over addition

(–2 × 3) × (–4) = –2 × (3 ×(–4)) = 24 Existence of multiplicative identity Any integer multiplied by 1 will result in the same integer.

For any three integers a, b and c, a × (b + c) = (a × b) + (a × c) This means that when we multiply an integer by the sum of two other integers, we get the same result by multiplying each addend separately and then adding the products together.

For example,

For example,

–4 × (3 + 2)= –4 × 5 = –20 and

a×1=a For example, a –4 × 1 = –4 b 17 × 1 = 17

(–4 × 3) + (–4 × 2) = (–12) + (–8) = –20 Existence of multiplicative inverse

Multiplication property of zero

Multiplication of an integer by –1

The product of any non-zero integer and its reciprocal is 1.

1 1 a × a = a × a = 1 or 1 1 (–a) × – = – × (–a) = 1 a a

For any integer a,

For any integer a,

For example,

Multiplicative inverse of (–12) is –

Chapter 3 • Integers

a×0=0

1 12

a × (–1) = –a

For example,

For example,

a –10 × 0 = 0

a 12 × (–1) = –12

b 5×0=0

b –25 × (–1) = 25

37

Example 1

What is the multiplicative inverse of −14 ?

Multiplicative inverse of –14 =

Example 2

Think and Tell

1 –14

What should be multiplied to 10 to get its additive inverse?

Why does the concept of the multiplicative inverse not apply to zero?

–1 should be multiplied with 10 to get its additive inverse.

10 × –1 = (–10) = additive inverse of 10 Do It Together

Fill in the blanks using the correct property of multiplication of integers.

23 × (13 + _____) = (23 × _____) + (_____ × 14)

a (–25) × (–8) = ( _____ ) × ( _____ )

c

b 52 × _____ = 52

d (17 × 5) × _____ = _____ × (5 × (–9))

Do It Yourself 1B 1

2

3

Evaluate the product. a 12 × 15

b –11 × 6

c –21 × 13

d –14 × (–6)

e 25 × 0

f 56 × (–1)

g –15 × (–24)

h 16 × (–8)

a –11 × 5 × (–2)

b –16 × 6 × (–18)

c 51 × (–26) × 6

d 12 × (–1) × (–8)

e 25 × (–12) × 3

f 56 × (–12) × 0

g –45 × (–4) × (–13)

h 11 × 0 × (–28)

Find the product.

Solve. a –23 × 15 × (–2) × 5 d

4

–6 × (–12) × –3 × (–10)

b –8 × (–16) × 5 × (–11)

c 13 × (–21) × (–26) × 6

e 34 × (–12) × 20 × 17

f –23 × (–24) × (–43) × 9

What will be the sign of the product if we multiply the integers a 5 negative integers and 5 positive integers. b 12 negative integers and 20 positive integers. c 20 negative integers and 12 positive integers. d 25 negative integers and 35 positive integers. e 33 negative integers and 50 positive integers.

5

38

Fill in the blanks with the correct property of multiplication. a 36 × 0 = ___________

b (–123) × ___________ = –123

c (–12) × (56) = ___________ × (–12)

d 56 × (–12+23) = ___________ × (–12) + 56 × ___________

e ___________ × (–1) = 2198

f (123 × 143) × (–36) = ___________ × 143 × (–36)

Word Problems 1

A submarine descends 55 feet per minute from sea

level. Where will be the submarine in relation to the sea level 8 minutes after it starts descending?

2

A shopkeeper earns a profit of ₹8 when selling a large packet of potato chips and a loss of ₹2 when selling a packet of biscuits.

a How much profit will he make on selling 25 packets of potato chips?

b How much loss will he make on selling 115 packets of biscuits?

3

A test has 25 questions. The test awards 5 points if the answer is correct and takes away 2 points if the answer is incorrect. A student answered 7 questions incorrectly. How many points did the student score?

Division of Integers You remember that the submarine travelled 14 km in 20 minutes. What if we need to find the distance travelled by the submarine in 10 minutes? This can be done with the help of division.

Division of two integers with unlike signs We already know that division facts come from multiplication facts. So if the Submarine travels –14 kilometres in 20 minutes, we need to find how far it would have travelled in half the time.

The above calculation can be represented as below sea level in 10 minutes.

(–14) = –7. This implies that the submarine travelled 7 km 2

Division of two integers with like signs Division of two integers with like signs produces the same resulting signs.

Dividing a positive integer by another positive integer results in a positive quotient.

Dividing a negative integer by another negative integer again results in a positive quotient.

Rules for division of integers can be given as: Rule 1: The quotient of two integers with unlike signs is always negative.

Rule 2: The quotient of two integers with like signs is always positive.

Chapter 3 • Integers

39

Example 1

Divide (–625) by 25.

(–625) = –25 (as Negative ÷ Positive = Negative) 25

Example 2

Ravi participated in a quiz competition where each incorrect answer deducts 3 points. If Ravi’s total negative score is –45, how many questions did he answer incorrectly?

Score of each incorrect answer: (–3) Total negative score: (–45) Number of question answered incorrectly: (–45) ÷ (–3) = 15 15 questions were answered incorrectly.

Do It Together

In a basketball match, Team A and Team B played four quarters. If Team A’s total score for the match was –48 points, what was their average score per quarter?

Number of quarters played = 4 Team A's total score for the match = __________ Average score per quarter = (–48) ÷ 4 = __________

Properties of Division of Integers Closure Property If a and b are two integers than

Commutative Property For any two integers a and b,

a ÷ b = c, where c may or may not be an integer.

For example, a. 25 ÷ 5 = 5 = integer

b. (–18) ÷ (5) = –3.6 = non-integer

Associative Property For any three integers a, b, and c,

a÷b≠b÷a

(a ÷ b) ÷ c ≠ a ÷ (b ÷ c)

This means that while dividing two integers a change in the order of the integers can change the answer.

This means that when dividing three or more integers, the order in which the divisions are performed changes the final results.

For example,

For example,

(−22) ÷ 11 = −2 but,

(625 ÷ 25)÷ 5=25÷5=5 but,

11 ÷ (−22) = −0.5

625 ÷ (25÷ 5)=625÷5=125 Division of an integer by itself While dividing a non-zero integer by itself, the result is always 1.

When dividing an integer by 1, the result is always the integer itself.

For any non-zero integer a,

For any integer a,

a÷a=1

40

Division of an integer by 1

a÷1=a

Division of an integer by –1

Example 1

Division of zero by an integer

When dividing an integer by –1, the result is the additive inverse of the integer.

While dividing zero by an integer, the result is always zero.

For any integer a,

For any integer a,

a ÷ (–1) = (–a)

Check whether the quotient of 482 ÷ 3 Is an integer.

Think and Tell

482 ÷ 3 = 160.66 which is not an integer.

Example 2

Can you think of special cases where the

Prove that the integers 256 and 16 are not commutative under division.

Commutative Property may hold true?

16 ÷ 256 = 1 16 1 As 16 ≠ , Hence, integers are not commutative under division 16 256 ÷ 16 = 16

Do It Together

but

Fill in the blanks. a (–125) ÷ (–25) = ________

b 0 ÷ 45 = ________

23 ÷ ________ = –23

d 136 ÷ ________ = 1

c

0 ÷ (a) = 0

Remember!

Dividing any non-zero integer by zero is undefined in mathematics and does not yield a valid result.

Do It Yourself 1C 1

2

Find the quotient. a 72 ÷ (–4)

b (–56) ÷ 7

c –176 ÷ (–11)

d 192 ÷ (–12)

e 344 ÷ 43

f (–984) ÷ 12

g (–676) ÷ (–26)

h –585 ÷ (–13)

a (–88 ÷ 4) ÷ –1

b 125 ÷ (125 ÷ 5)

c (192 ÷ (–16)) ÷ 4

d 324 ÷ (–18 ÷ 2)

e (900 ÷ (–18)) ÷ 6

f (–3060 ÷ (–36)) ÷ 5

g 0 ÷ (–1508 ÷ 29)

h (1000 ÷ (–20)) ÷ 100

Solve.

3

Find the number which when divided by (–1) gives 145.

4

State True or False. a Division is commutative for integers.

__________________

b Division has the associative property for integers.

__________________

c The quotient of two negative integers is always negative.

__________________

d Division by zero is defined for integers.

__________________

e The quotient of zero divided by any non-zero integer is zero.

__________________

5

Give two pairs of integers such that a ÷ b = –15

6

Find the value of (12 x –3) ÷ (–2) + (–4)

Chapter 3 • Integers

41

Math in Action 1

The temperature recorded at 12 noon was 18° C above zero. If it falls at the rate of 3° C per

hour until midnight, at what time will the temperature be 6° C below zero? What will be the temperature at midnight?

2

In the first level of a video game, 2 points are deducted for each time a character falls. In the

second level, each fall deducts 4 points. Rahul lost 10 points in the first level and 16 points in the second level. In which level did Rahul character fall more often?

Points to Remember Addition

Like sign: Add the absolute values and place the common sign.

Subtraction

Unlike sign: Find the difference between the absolute values and use the sign of the integer with the greater absolute value.

To subtract an integer b from another integer a, add the additive inverse of b to a.

Integers

{…..–3, –2, –1, 0, 1, 2, 3….}

Multiplication

Like Sign: The product of two integers with like sign is positive.

Unlike Sign: The product of two integers with unlike signs is negative.

42

Division

Like Sign: The quotient of two integers with like sign is positive.

Unlike Sign: The quotient of two integers with unlike signs is negative.

Math Lab Integer Operations Relay Setting: In groups of 5 Materials Required: Index cards with different integer expressions written on them, stopwatch or timer.

Method:

23 – 52 =? = ? +23 – 52

15 × (–5) = ?

96 =?

–4

(–15) + (–9) = ?

1

Place the index cards face down on a table. Start the stopwatch or timer and begin the relay.

2

The first team member will pick up an index card, read the expression aloud, solve it, and return it

3

The relay continues until all the index cards have been used or until a designated time limit is reached.

4

The team with highest number of correct solutions wins the game!

96 � (–4) = ?

to their team to tag the next person.

Chapter Checkup 1

2

3

4

Solve. a (–256) + 362

b 214 – (–126)

e (–659) + (-1352)

f (–2698) – (–1236)

c 652 + (–129)

d 248 – (–369)

c (-36) x 48

d (245) x (-57)

c 5525 by (–25)

d (-5472) by 36

Find the product. a 25 x (-89)

b (-125) x (-26)

e (1245) x (-142)

f (-2365) x (123)

Divide. a 363 by (-11)

b (-2652) by (-51)

e 2852 by 23

f (-9594) by (-41)

Fill in the blanks. a (–156) + 389 = ______

b 524 + ______ = 256

c ______ – (–269) = –698

d 29 × ______ = (–1885)

e ______ � (–17) = 56

f –568 – (–258) = ______

g ______ × (–15) = –5475

h (2356) – ______ = 5886

i

______ + (–1265) = –4526

j

k (–1265) × (–352) = ______

l

(–3562) � (–137)= ______

3654 � ______ = (–87)

Chapter 3 • Integers

43

5

6

Simplify. a 42 ÷ (–6 + 5)

b –64 ÷ 4 × (2 – 6)

e 7 × (5 + 3) ÷ 4 × (9 – 2)

f (6 + 2) – 15 ÷ (5 × 2)

d 4(–12 + 6) ÷ 3

c –148 + (–157)

d 365 + 124

Solve and write the additive and multiplicative inverse. a 25 + (–63)

7

c (9 ÷ 3) + 7 × (4 ÷ 2)

b –125 + 47

State True or False. a The associative property of multiplication states that changing the grouping of the numbers being multiplied does not change the product.

b The additive inverse property states that every integer has an additive

inverse such that the sum of the integer and its additive inverse is zero.

c The multiplicative inverse property states that every non-zero integer has a multiplicative inverse such that the product of the integer and its multiplicative inverse is one.

________________ ________________ ________________

d The commutative property holds for division of integers.

________________

e The associative property holds for subtraction of integers.

________________

8

Add the sum of (135) and (-325) to the difference of 253 and (-528).

9

What should be multiplied to (-165) to get a product of 9240?

10 Add the product of 25 and (-36) to the quotient of 2380 divided by (-68).

11 What should be divided from 23072 to get (-412)?

12 An aeroplane is flying at 12,000 feet. The plane climbs 15,000 feet to approach cruising altitude. After a few minutes at this new altitude, the plane hits turbulence and descends 23,945 feet. Express each increase and decrease in altitude as an integer operation and determine the new altitude of the plane

Word Problems

44

1

A mercury thermometer records a temperature of 6℉ at 11 a.m. If the temperature drops by 3℉ every hour, what will be the temperature by 4 p.m of the same day.

2

Kajal gets into the elevator on the fourth floor of a shopping mall. She goes up 10 floors to reach the food court. After an hour, she went down 5 floors to buy books. At which floor is she now?

3

A shoe store marked ₹55 off on the price of each pair of shoes in stock. If the total reduction in price is ₹7,590, what is the total number of pairs of shoes in the store?

4

Measures of Central Tendency

Let’s Recall Ritika wanted a pet for herself and asked her mother if she could get one. She collected the data of number of pets in her neighborhood and presented it to her mother.

Number of Pets

Number of Households

0

15

1

12

2

5

3

2

4

1

How many households have no pets?

15 households have no pets.

How many households have more than 1 pet?

Households with 2 pets + Households with 3 pets + Households with 4 pets =5+2+1 =8 So, 8 households have more than 1 pet.

Let’s Warm-up Read the data given above and fill in the blanks.

1

________ households have 3 pets.

2

________ households have less than 2 pets.

3

The total number of pets in the neighborhood is ____________.

4

There are _______ households with 1 pet each.

5

The difference between the households which have pets and which do not have pets is _______.

I scored _________ out of 5.

Mean, Mean,Median Medianand andMode Mode Real Life Connect

Ria’s parents are planning to go on a vacation to Jaipur for the first time.

Ria: Papa! What is the weather in Jaipur like?

Ria’s father checks the online weather forecast for Jaipur for the next 15 days and finds the following: Days

Day 1

Day 2

Day 3

Day 4

Day 5

Day 6

Day 7

Day 8

Day 9

Day 10

Day 11

Day 12

Day 13

Day 14

Day 15

Temp (in °C)

32

34

32

37

42

40

30

32

36

42

44

44

42

41

42

Ria: Papa! The temperature is so different every day. What clothes should we pack?

Arithmetic Mean or Mean One possibility is that Ria and her father want to take a decision based on a single value that represents the entire data. But what will that value be and how is it calculated?

Arithmetic mean or mean or average of a data is a value that can represent all of the data. It is defined as the sum of all the observations of data divided by the total number of observations.

Mean =

Sum of all observations No. of observations

But data could be presented in different forms. How do we calculate the mean in all such cases?

Let us learn!

Mean of Ungrouped Data In the last chapter, we learnt that when data is listed as individual values or observations it is called ungrouped data.

For example, the temperature forecast data that Ria’s father showed is also ungrouped data.

So, in the above case, the average temperature can be calculated by dividing the sum of all the observations (temperature in °C) by the total number of observations (the number of days). Thus, the mean temperature can be calculated as: Mean temperature = 32 + 34 + 32 + 37 + 42 + 40 + 30 + 32 + 36 + 42 + 44 + 44 + 42 + 41 + 42 °C 15 =

570 °C 15

= 38 °C So, the average or mean temperature is 38 °C.

Remember!

The mean of the data may not always be an observation from the data.

Therefore, if Ria and her father want to make a decision based on a single temperature that reflects the overall duration of their stay, they should pack their clothes based on the mean temperature, 38 °C. 46

Example 1

Find the mean of the first 5 natural numbers.

Step 1

Step 2

Write the first 5 natural numbers.

Find the sum of the observations.

1, 2, 3, 4 and 5.

1 + 2 + 3 + 4 + 5 = 15

Step 3 Count the total number of observations. Here, they are 5.

Think and Tell

Step 4

Could the mean be more than the

Find the mean using the formula.

Mean = =

highest value of the data?

Sum of all observations No. of observations 15 =3 5

So, the mean of the first five natural numbers is 3.

Example 2

The following table shows the number of goals scored by three lead players in four matches.

Players

Vishal

Bharat

Tejas

Goals in Match 1

4

0

4

Goals in Match 2

5

2

6

Goals in Match 3

2

6

Did not play

Goals in Match 4

5

4

8

Which player scored the highest average goals?

We know that, mean = Sum of all observations No. of observations Let’s first arrange the data.

Players

Vishal

Bharat

Tejas

Goals in Match 1

4

0

4

Goals in Match 2

5

2

6

Goals in Match 3

2

6

Did not play

Goals in Match 4

5

4

8

Total Goals Scored

16

12

18

Total No. of Matches - Player Wise

4

4

3

Calculating Averages - Player Wise

16 4

12 4

18 3

Average Goals - Player Wise

4

3

6

Thus, Tejas has the highest average score of 6 goals.

Chapter 4 • Measures of Central Tendency

47

Do It Together

Find the mean of the first five prime numbers.

Step 1 Find the first five prime numbers.

2, 3, _______, _______, _______

Did You Know?

Step 2

Prof. Prasanta Chandra Mahalanobis

Find their sum.

is known as the father of modern

2 + 3 + _______ + _______ + _______ = _______

statistics in India. In 1968, he

Step 3

second highest civilian award

was honoured with India’s - the Padma Vibhushan

Find the mean.

Mean =

5

= _______

Thus, the mean of the first five prime numbers is _______.

Mean of Grouped Data We learnt that grouped data is a way of organising data by putting observations into groups. Instead of looking at each number individually, we group them together based on their values.

We saw that the temperature data was ungrouped. Let us try to group it and find the mean.

Step 1 Create a grouped data table. The observation column is denoted by x, while the frequency by f.

Temperature (in °C) (x)

Tally Marks

No. of Days (frequency)

30

|

1

32

|||

3

34

|

1

36

|

1

37

|

1

40

|

1

41

|

1

42

||||

4

44

||

2

Total We know that, mean =

48

(f)

15 Sum of all observations No. of observations

Step 2 Find the sum of all observations.

First, multiply each observation by its frequency and list in a column as shown and then find the sum.

Temperature (in °C) (x)

Tally Marks

No. of Days (Frequency)

|

1

30

3

96

1

34

1

36

1

37

1

40

1

41

4

168

2

88

15

570

30

|

34

|

36

|

37

|

40 41 42

(f)

|||

32

| |||| ||

44

f×x

Total

Sum of all observations = 570 We can in general represent this sum of all

observations by ∑( f × x). The symbol Σ (called sigma) is a Greek letter that represents summation.

Step 3 Find the total number of observations as the sum of all the frequencies.

We see from the table that the total number of observations is 15.

In general, the total observations can be denoted as ∑f.

Error Alert!

Step 4

To find the total number of observations, NEVER add all the observations. We ALWAYS add their frequencies.

Calculate the mean.

Mean =

Sum of all observations No. of observations

=

∑( f × x) ∑f

=

570 15

= 38 °C

The mean is 38 °C. Example 3

Data on the number of books read by students in a class is represented in the frequency distribution table below: No. of Books

1

2

3

4

5

No. of Students

6

3

5

3

1

Find the mean of the data.

We know that, mean = Sum of all observations = No. of observations

∑( f × x) ∑f

Remember!

The summation sign is especially helpful in case of large data. There is no need to write each observation separately.

Let us find ∑f and ∑( f × x).

No. of Books (x)

No. of Students (f)

f×x

1

6

6

2

3

6

3

5

15

4

3

12

5

1

5

Total

Σf =18

Σ( f × x) = 44

Chapter 4 • Measures of Central Tendency

Thus, the mean =

Σ( f × x) 44 = = 2.44 Σf 18 49

Example 4

The table below shows the pocket money of 20 students.

Pocket Money

₹100

₹120

₹105

₹125

₹140

No. of Students

4

5

7

3

1

Pocket Money (x)

No. of Students ( f )

f×x

₹100

4

400

₹120

5

600

₹105

7

735

₹125

3

375

₹140

1

140

Total

∑f = 20

∑( f × x) = 2250

Find the mean pocket money.

Let us find ∑f and ∑( f × x).

∑( f × x) 2250 = = 112.5 Σf 20 Thus, the mean pocket money is ₹112.50.

Mean =

Example 5

The mean of x, x + 1, x + 3, x + 5, x + 7 is 228.2. What is the value of x?

Mean = 228.2 =

Sum of all observations Total number of observations

x + (x + 1) + (x + 3) + (x + 5) + (x + 7) 5

228.2 × 5 = 5x + 16

5x + 16 = 1141 or 5x = 1141 − 16 5x = 1125 x = 225 Do It Together

Find the arithmetic mean or the average of the marks (out of 10) in a class test.

Marks (out of 10)

5

4

6

2

7

8

9

No. of Students

2

3

5

7

4

5

4

Step 1

Step 2

Let us find ∑f and ∑( f × x).

Find the mean.

Marks (x)

No. of Students ( f )

f×x

2

7

_______

5

2

_______

4

28

4 6 7 8 9

Total

50

3 5

12

_______

5

_______

∑f = 30

∑( f × x) = _______

4

_______

∑( f × x) = = _________ Σf 30 Thus, the arithmetic mean or the average marks of the students is __________.

Mean =

Do It Yourself 4A 1

Find the mean of the first 10 odd numbers.

2

The temperatures in a city were recorded as 3 °C, –7 °C, –1 °C, –5 °C, 2 °C. Find the mean temperature for the

3

The following table shows the number of pages having a certain number of illustrations.

recorded data.

No. of Illustrations (per page)

2

4

5

6

No. of Pages

25

30

45

12

Find the mean number of illustrations per page.

4

The mean of 5 numbers is 26. One of observations, which was 6, was removed. Find the new mean. Is the new

5

The mean of 40, 50, 55, a and 58 is 56.2. What is the value of a?

6

The mean of x, x + 2, x + 3, x + 5, x + 7 is 92. What is the value of x?

7

The mean of 30, 25, x, 50, y, 45, z is 40 and the sum of y and z is 76. What is the value of x?

mean greater or less than the original mean?

Word Problems 1

A group of friends went to a restaurant and shared the bill equally among themselves. The total bill

2

A group of soldiers went on a journey. They had to take road trip through India which passes

came to ₹2,080. If there were 8 friends in the group, what was the amount each friend had to pay? through many rivers along the way. The rivers were listed and the lengths of the rivers (in kilometres) is given.

River

Tapi

Yamuna

Cauvery

Ganga

Length (in km)

724

1376

800

2525

Find the mean length of the rivers visited by the troop.

3

The mean attendance of a class from Monday to Saturday was 33. If the mean attendance

from Monday to Thursday was 32 and that from Thursday to Saturday was 34, then find the attendance on Thursday.

Mode Real Life Connect

We learnt that the average can represent the whole data. But what if Ria’s father wants to take a decision based on the temperature that occurs most frequently?

In this case, Ria and her father must choose what is called the ‘’Mode’’ of a data.

A value in the data that occurs most frequently in a given data is called Mode.

Just like average, mode can also be calculated for both ungrouped and grouped data. Let us understand how.

Chapter 4 • Measures of Central Tendency

51

Special Case 1: More than one mode It is possible that in a set of data there are several observations with the same highest frequency (more than 1). In such a case, all such observations are the modes.

For the following data: 2, 2, 3, 4, 4, 5, 6, 6, 7, 7 The modes are → 2, 4, 6 and 7 Special Case 2: No defined modes If all the values in a set of data occur exactly once, there is no value that can be considered the mode. In this case, the data is said to have no mode.

For the following data: 1, 2, 3, 4, 6, 7 The mode does not exist because each observation occurs exactly once.

Mode of Ungrouped Data Let’s once again consider the data gathered by Ria’s father when they were planning to go on a vacation.

Days

Day 1

Day 2

Day 3

Day 4

Day 5

Day 6

Day 7

Day 8

Day 9

Day 10

Day 11

Day 12

Day 13

Day 14

Day 15

Temp (in °C)

32

34

32

37

42

40

30

32

36

42

44

44

42

41

42

Let’s find the mode of the above data.

Step 1 Arrange the temperatures (in °C) in ascending order.

Remember!

30, 32, 32, 32, 34, 36, 37, 40, 41, 42, 42, 42, 42, 44, 44

Mode will always be an observation from the given data.

Step 2 Find the observation that occurs most frequently.

30 °C, 32 °C, 32 °C, 32 °C, 34 °C, 36 °C, 37 °C, 40 °C, 41 °C, 42 °C, 42 °C, 42 °C, 42 °C, 44 °C, 44 °C Occurs thrice

Occurs 4 times

Occurs twice

We see that the observation 42 °C occurs the most, i.e., 4 times. Thus, it is the mode of the data.

So, Ria and her father can pack their clothes for the most frequent temperature or the mode of the data, 42 °C. Example 6

The marks scored (out of 20) by Vikas in various class tests during a term are given below: 18, 15, 17, 13, 17, 19, 17, 16, 14 Find the mode.

Step 1:

Step 2

Arrange the marks in ascending order.

Find the observation that occurs most frequently.

13, 14, 15, 16, 17, 17, 17, 18, 19

13, 14, 15, 16, 17, 17, 17, 18, 19 Occurs thrice

Thus, the observation 17 occurs the most, i.e., 3 times.

Thus, it is the mode of the data.

52

Do It Together

The amount spent by Raj in 10 days on commuting to office is given below: ₹120, ₹130, ₹130, ₹135, ₹120, ₹140, ₹130, ₹250, ₹200, ₹120 Find the mode of the of data.

Step 1 Arrange the observations in ascending order.

₹120, ₹_____, ₹_____, ₹130, ₹_____, ₹_____, ₹135, ₹_____, ₹_____, ₹250 Step 2 Find the observation that occurs most frequently.

The observation(s) __________________ occur(s) three times each. Thus, the mode(s) of the data is/are ₹__________________.

Mode of Grouped Data Remember, we converted the above data in the grouped form to find the mean. We obtained the following table.

Temperature (in °C)

Tally Marks

No. of Days (frequency)

30

|

1

32 34 36 37 40 41

||| | | | | |

3 1 1 1 1 1

42

||||

4

44

||

2

Error Alert!

Temperature

Frequency

°C Error42Alert!

5

43 °C

Mode is 5

3 Mode is 42 °C

The observation with the highest frequency is the mode of the data.

From the table, the observation 42 °C has a frequency of 4, which is more than any other observation in the data. Thus, the mode of the data is 42 °C. Example 7

The following table shows the colour of dresses worn by various guests at a party.

Colour of Dresses

No. of Guests

Black

2

Red

9

Blue

4

Brown

7

Pink

4

Find the modal colour of the dresses worn by the guests.

From the table it is clear that ‘Red’ is the modal colour as it occurs most frequently.

Chapter 4 • Measures of Central Tendency

53

Do It Together

Vivaan wrote a few numbers on a piece of paper. The most frequent number (i.e., mode) was 8.

One number got wiped out. Can you find out what that number was?

4, 6, 6, 7, 8, 3, 8, 6, 5, 8, ?

Let us start by arranging the data in ascending order 3, 4, 5, 6, 6, 6, 7, 8, 8, 8, ?

Can the missing number be 3? No, because then the modes will be 8 and 6.

Can the missing number be 4? No, because then the modes will be 8 and 6.

Can the missing number be 5? _________, because then the mode(s) will be _________.

Can the missing number be 6? _________, because then the mode(s) will be _________.

Can the missing number be 7? _________, because then the mode(s) will be _________.

Can the missing number be 8? _________, because then the mode will be _________.

Do It Yourself 4B 1

State true or false. a The mode is the observation that occurs the least number of times in a data set. _________ b The mode is always one of the numbers in the data. _________ c The data 2, 4, 3, 4, 6, 3, 6, 7, 4 has mode 3. _________ d There is only one mode for the given data. _________

2

Find the mode of the following years of experience of teachers in a school: 12, 5, 7, 10, 3, 2, 1 ,7, 6, 7, 5, 10, 7, 4, 2, 1, 4, 4

3

The following data shows the marks obtained (out of 20) by 14 students of a class in a test.

Name

Jai

Ajay

Ron

Aru

Ria

Anya

Sia

Shiv

Remo

Dev

Dia

Dino

Sam

Pihu

Marks Obtained

12

14

13

16

18

15

18

19

12

14

16

15

18

11

Find the mode of the data. How many students got the highest marks?

4

In a singing competition, points were awarded to 15 participants with a maximum possible score of 60. The allotted points for each participant were as follows: 60, 52, 42, 40, 43, 44, 46, 45, 42, 41, 40, 46, 48, and 46.

Determine the mode of the given data.

5

The number of books read by 30 students of a class is given below: No. of Books

1

2

3

4

5

6

7

8

No. of Students

2

3

7

8

6

0

2

2

Find the mode of books read by the students.

54

Word Problem LIBRARY

1

The librarian keeps track of the types of books that children choose from the library. She notices that the last 10 books belonged to the following genres: Fiction, Poetry, Drama,

Drama, Fiction, Fantasy, Poetry, Drama, Fantasy, Drama. Find the modal genre.

Median Real Life Connect

We saw how Ria and her father decided to pack clothes based on:

a. A single value for the temperature that represents the whole data set, i.e., the arithmetic mean or mean, which was 38 °C. b. A single value of temperature that occurred most frequently, i.e., the mode, which was 42 °C.

What if Ria and her father wanted to take a ‘balanced’ decision to choose a temperature value that falls exactly in the middle of the temperature extremes? So, they would expect exactly as many hotter days as there were cooler days.

The median is the middle value in a set of data when arranged in ascending or descending order.

Let’s find the median of the data collected by Ria’s father.

We arrange the data in ascending or descending order.

30, 32, 32, 32, 34, 36, 37, 40, 41, 42, 42, 42, 42, 44, 44 Values less than 40

Median

Values more than 40

Here, 40, i.e., the 8th value, has exactly 7 values higher and 7 values lower. Therefore, 40°C is the median of the data.

Let’s learn more about median calculation.

Case I: When the total number of observations is an odd number, When the total number of observations is odd, the median would be the observation in the middle, n + 1 th i.e.,   observation in the arranged data. Let us consider the following data.  2 

20

15

4

10

17

8

1

Step 1

Error Alert!

Arrange the data in ascending order.

1

4

8

10

15

Step 2 Find the total number of observations.

Total number of observations is 7.

Chapter 4 • Measures of Central Tendency

17

20

 n 1 th, you find the +  2 

By using the 

median place and NOT the median itself. The observation in that place is the median.

55

Step 3 n + 1th observation in the arranged data is the median.

The    2 

Remember!

7 + 1th observation, which is the 4th observation is the median.   2 

Here 

When arranged in ascending or descending order, the median has half of the observations lying to its left and the other half to its right.

Step 4 Find the median.

1

4

8

10

Less than median

15

Median

17

20

More than median

Thus, the median value is 10. Exactly three values are more than 10 and three values are less than 10.

Case II: When the total number of observations is even Let us consider the same data as above, but just add another observation to it. So, the new data is: 20

15

4

10

17

8

1

22 New observation

Step 1

Step 2

Arrange the data in ascending order.

Note the total number of observations.

1

4

8

10

15

17

20

22

Step 3

Step 4

Find the two middle observations.

1

4

8

10

15

17

Total number of observations is 8.

20

22

Two middle observations

 n th n th These are the   and the  + 1 observations. 2 2 

The median is the average of the middle two observations.

Median = Median is

1   n th  n th    +  + 1  observation 2  2 2   (10 + 15) 2

= 12.5

Here, these are the 4th and 5th observations.

Example 8

Find the median of the following data: 10, 8, 14, 16, 10, 6, 18, 14, 16 Here, n = 9, which is an odd number.

By arranging the data in ascending order, we get:

Think and Tell

6, 8, 10, 10, 14, 14, 16, 16, 18

Is it necessary that the median will

Middle value

always be present in data?

Thus, the median is 14.

We can also use the formula to find the median.

th th th Median =  n + 1  observation =  9 + 1  observation =  10  observation = 5th observation  2   2  2 th The 5 observation is 14. Thus, the median is 14.

56

Find the median of the first 10 prime numbers.

Example 9

We know that the first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 Since they are already arranged in ascending order, we can start finding the median.

Ascending order of observations: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 Here, the number of observations, n = 10, is even.

th th So, here the median is the average or mean of  n  and  n + 1 observations. 2 2 

That is, n = 10 = 5th observation, and  n + 1 = 10 + 1 = 5 + 1 = 6th observation.

2 2 2  2 The ascending order of observations is: 2, 3, 5, 7, 11, 13 ,17, 19, 23, 29 5th and 6th observations

th  th Thus, Median = 1 n + n + 1  observations = 1 {5th + 6th} observations = 1 (11 + 13) = 1 × 24 = 12 2  2  2   2 2 2 Thus, the median of first 10 prime numbers is 12.

Example 10

The median of the given data is 50. What is the value of a?

15, 20, 25, a, 80, 120 Total observations = 6 (Even number) 3rd term + 4th term 2 25 + a = 50 × 2

Median =

50 =

25 + a 2

a = 100 − 25 = 75 Do It Together

The heights of 7 girls of a class are given in the following table: Name

Radha

Avni

Rita

Jaya

Shikha

Shilpi

Rinki

Height (in cm)

160

164

148

169

171

154

162

Find the median height.

To find the median height (in cm): Step 1

Step 2

Arrange the heights (in cm) in ascending order.

Find the total number of observations.

148, _____, _____, _____, _____, _____, 171

The total number of observation = 7, which is an

Step 3

_________ (even/odd) number.

Note the middle observation(s) as _________ using the formulae.

The median height is _________ cm.

Chapter 4 • Measures of Central Tendency

57

Do It Yourself 4C 1

Fill in the blanks. a To find the median, the data can be arranged in _______________ or ____________ order.

b If the number of observations is an odd number, then the median is the ____________ observation. c The median of first 6 odd numbers is _________.

2

3

The enrolment in a school during six consecutive years is given below. Find the median of the data.

Year

2018

2019

2020

2021

2022

2023

Enrolments

1555

1670

1850

1780

1400

2540

The runs scored by 11 players of a school cricket team are given in the table below.

Players Ishan Ravi Murli John Iqbal Arun Vikas Viru Vipin Jai Aman Runs

60

15

120

40

80

54

15

25

62

6

14

Find the median score of the team.

4

The time taken (in minutes) by students in a class to complete a certain task is given below: 7, 21, 12, 15, 18, 9 a Find the median time taken by the students. b If the largest time value is replaced by 35, will it affect the median? Why? c Can new students be added to the above data, so that the median does not change?

5 6

The numbers 2, 4, 6, 6 , 2x + 1, 9, 9, 10, 11 are written in ascending order. If the median of the given data is 7,

find the value of x.

Consider the data arranged in ascending order.

10, 20, x , 30, 40, 50, 70 If the median of the data is 30, then what is the maximum possible value of x?

7

Consider the data arranged in descending order.

60, 55, 54, 53, 45, 35, x , 25, 20, 17, 15, 10 a If the median of the data is 30, then what is the value of x? b If 60 is replaced by 20 then what is the new median?

Word Problems 1

A group of friends went to a store to buy snacks. They decided to pool their money to make a purchase. Each friend contributed a different amount of money. The amounts contributed by each friend are as follows: ₹15, ₹20, ₹10, ₹25, ₹30 Find the median amount contributed by the group of friends.

2

58

The front row in a movie theatre has 23 seats. Rohan booked a seat in the front row that occupied the median position. Which seat did Rohan book?

Choosing the Right Measure of Central Tendency From the previous sections, we learnt about the different ways in which Ria and her father could have taken the decision about their packing. The mean, median and mode helped them decide. These are called the measures of central tendency.

Measures of central tendency are ways to find the typical or “central” value of data.

Mean: The mean, also called the average, is a value that can represent the whole data.

Median: The median is the middle observation in the data when arranged in an order.

Mode: The mode is the observation that appears most frequently in the data.

But which measure of central tendency do we choose and why?

We can choose the appropriate measure of central tendency according to the purpose of our exercise. The following table highlights when each measure of central tendency should be used.

Mean When to Use

• to find one value that represents the whole data.

Median

Mode • to recognise the most commonly occurring observation.

• to find a “middle” value in data that has a sizeable number of observations.

• to see which observation divides the entire set of data into two equal halves or parts.

Example

Example 11

Sachin Tendulkar’s average in One-Day International Cricket matches is 44.8. He played 463 matches. This means that if we replace all his scores in individual matches with this average, it would represent his performance throughout his career.

In an exam, if the median score was 75%, this implies that exactly half the number of students who participated scored below 75% and the other half scored above 75%

A shopkeeper sells T-shirts in 5 sizes: Extra Small, Small, Medium, Large and Extra Large. The most frequently sold T-shirts represents the modal size and is most important for his business.

A researcher wants to study the duration of the commute of employees in a city. The researcher randomly selects 15 employees and records the duration of their commute in minutes. The dataset consists of the following durations of commute: 25, 30, 33, 40, 42, 50, 53, 60, 62, 70, 74, 78, 85, 90, 93. Which measure of central tendency should the researcher use to describe the average duration of their commute?

Using Mean: As the researcher wants to describe the average duration of their commute, the mean can be given as: Mean = (25 + 30 + 33 + …….. + 93) = 885 = 59 15 15 Using Median:

Remember!

One or more measures could possibly be good measures.

Another way of describing the average duration of their commute is by using median. It can be given as:

th th Median = As the data has an odd number of observations, the median will be  n + 1  observation = 15 + 1  2   2  = 8th observation.

Chapter 4 • Measures of Central Tendency

59

Hence, the median for (25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95) is 60.

Middle value

Using Mode:

We know that mode is the most commonly occurring observation but in the above case, none of the data is repeating. Therefore, mode cannot be used to describe the average duration of their commute for the above data.

Do It Together

Which measure of central tendency should the retailer use to plan his purchase for the upcoming week, considering the sales record of shirts in five sizes during the first week of the month?

Shirt Size No. of shirts sold

Extra Small

Small

Medium

Large

Extra Large

70

92

104

68

45

Using Mean and Median: As the retailer is dealing with _________ (categorical/numerical) data, he _________ (can/can't) use mean or median to plan his purchase for the upcoming week.

Using Mode: As mode is used to recognise the _________ (most/least) commonly occurring observation, the retailer can use mode to plan his upcoming purchase.

The mode for the given data is _________ .

Therefore, only _________ can be used to plan the purchase for the upcoming week.

Do It Yourself 4D 1

Fill in the blanks. a The measures of central tendency are __________, __________ and __________. b __________ represents one mathematical equivalent for the data. c __________ is used when you need a “middle” value of a data set that has a sizeable number of observations. d __________ cannot be defined for data with no repeated values.

2

A researcher is studying the heights of trees in a forest. The heights of the trees are as follows (in metres): 5, 7,

3

A charity organisation collected donations from individuals for a fundraising event. The donations are as follows:

10, 12, 15, 18, 22, 30. Which measure of central tendency should not be used by the researcher and why?

₹10, ₹25, ₹50, ₹100, ₹250, ₹500, ₹1,000, ₹2,000, ₹5,000, ₹10,000. Which measure of central tendency should be used to represent the typical donation, and why?

4

The scores of the team members of a golf team are shown below.

72, 73, 72, 76, 72, 74, 73, 75, 76, 74, 72 Which central tendency of measure would be the most appropriate to find the mean score of the team member?

5 60

A data set has 50 different values. 47 values are between 1 and 5 while the rest 3 values are more than 50. Which measure of central tendency will better represent the typical value of the data set?

Word Problem 1

The teacher took a surprize test. The marks scored (out of 100) by Ravi and his classmates is shown.

87

75

64

89

0

78

93

95

90

84

Calculate the mean score. Calculate the median score of the class. Explain which of the measures of central tendency better explains the score of the class. Why?

Points to Remember • •

Mean, median and mode are the three measures of central tendency.

Mean:

Sum of all observations No. of Observations  Cannot be more than the highest value of the data.  Mean =

•

 Need not always be an observation from the data.

Median:

 The middle value of a set of data when the data is arranged in ascending or descending order.

n + 1th observation.   2   Even number of observations, the median is given by the average of the two observations in the th th middle, i.e.,  n  and  n + 1 observations. 2 2  Mode:  Odd number of observations, the median is given by the 

•

 That occurs the greatest number of times in the data.  Will always be an observation from the given data.

 A set of data can have more than one mode, or even no mode.  Is used when the data has many identical values.

Math Lab ShoeTopia Setting: Groups of 6 Materials Required: Chart paper, markers, and sticky notes Method: 1 Divide the class into groups of 6 each. Give each group a sticky note. Each child in the group notes down her/his shoe size on the sticky note.

Chapter 4 • Measures of Central Tendency

61

2 Once all the groups are done, the teacher asks the students to find the following for their respective groups:

a The mean shoe size

b The modal shoe size

c The median shoe size

d The biggest shoe size

e The smallest shoe size 3 The group that finds all the correct answers first, wins.

Chapter Checkup 1

For each set of data, find the mean, median and mode.

2

Find the mean and median of:

3

The weights of 5 students are given as 38 kg, 41 kg, 36 kg, 39 kg, and 41 kg. Find the mean weight recorded in

4

Find the median of the data: 3, 2, 4, 2, 5, 6, 2, 7, 3, 4, 2

5

The marks obtained (out of 10) by the students in a class test are given below:

a 8, 12, 16, 13, 9, 10, 15, 12

b 12, -15, -32, -11, -15, 14, -30, -18

a the first 10 natural numbers

b the first 8 prime numbers

c the factors of 36

the data.

4, 6, 7, 5, 3, 5, 4, 5, 2, 6, 2, 5, 1, 9, 6, 5, 8, 4, 6, 7 Find the mean and median of the data using a frequency distribution table.

6

Find the mode of the following data: 20, 26, 16, 22, 20, 30, 22, 20, 26

7

The runs scored by 11 players in a cricket match are given below: 6, 15, 120, 50, 100, 80, 20, 15, 8, 10, 15, 90, 90 Prepare a frequency distribution table and find the mode.

8

The weight of 70 students is given below: Weight (kg)

50

51

54

57

58

60

61

No. of students

10

11

16

14

9

7

3

Find the mean, median, and mode of the given data.

9

The recorded rainfall values (in mm) for each of the last 10 days Meghalaya during the month of June 2023 was: 97

96.5

125.5

52.2

82.1

108.5

35.6

13.6

1.5

2.7

Which measure of central tendency would be most suitable and why?

62

in

10 A survey asked participants to rate a product on a scale of 1 to 10. The ratings received were as follows: 4, 6, 7, 8, 9, 9, 10. Which measure of central tendency should be used to represent the typical rating, and why?

11 What is the median age of a family whose members are 45, 42, 38, 35, 18, and 10 years old?

12 The size of 10 shirts are as follows: 80, 85, 90, 80, 80, 85, 85, 90, 80, 95 Find the modal size. If the size of one shirt is misread as 80 instead of 85, find the correct modal size.

13 Find the mean and median of the following set showing the number of hours of operating life of 20 flashlight batteries:

19, 25, 20, 21, 25, 19, 21, 22, 24, 20, 25, 18, 23, 21, 24, 20, 19, 23, 25, 18 14 The weights (in kg) of 15 students are as follows: 27, 30, 42, 43, 36, 34, 35, 37, 28, 29, 31, 44, 41, 32, 33.

If 30 kg weight is replaced by 25 kg and 41 kg by 45 kg, then find the new median.

15 The ages of a group of students are as follows: 12, 14, 15, 15, 16, 17, 18, 18, 19, 21. Which measure of central tendency should be used to represent the typical age of the students, and why?

Word Problems 1

The teacher of class 7 found the mean weight of a class consisting 20 students as 45 kg. Later on two more students Ankit and Suhani weighing 55 kg and 52 kg, respectively, join the class. What is the mean weight of the class now?

2

The ages of a group of people are 22, 28, 30, 29, and 31. If another person, aged 32, joins the group, will the median age change? If yes, what will the new median age be?

3

Mohan secured 75, 82, 79 and 76 marks in four tests. What is the lowest number of marks he can secure in his next test, if he has to have a mean score of 80 marks in five tests.

Chapter 4 • Measures of Central Tendency

63

5

Data Handling

Let’s Recall We can collect data through various methods and organise it in tabular form. Let’s consider an example.

Roy collected the data for the most international test runs by players.

He put the data in a tabular form. a

b

Player Name

Runs

Which player scored the most runs?

Sachin Tendulkar

15,921

Sachin Tendulkar scored the most runs.

Ricky Ponting

13,378

Jacques Kallis

13,289

Rahul Dravid

13,288

Alastair Cook

12,472

Which players scored 13,288 runs?

Rahul Dravid scored 13,288 runs.

Let’s Warm-up Study the data given above and fill in the blanks.

1

_______________ scored 1 run more than Rahul Dravid.

2

The total runs made by Ricky Ponting and Alastair Cook are _______________.

3

The total runs scored by the 5 batsmen are _______________.

4

_______________ scored the least runs of the 5 players.

5

Sachin Tendulkar scored _______________ more runs than Jacques Kallis.

I scored ____________ out of 5.

Organisation and Representation of Data Real Life Connect

Annie owned a warehouse. She kept a stock of multiple products at her warehouse. She collected the data of all the products, including the stock available and the sale of the products.

Remember!

Numerical information or facts collected are known as data.

Data Organisation Annie collected the data for all the categories of products and different types of products available at her warehouse.

Category

Dairy

Number of Products

11

Fruit and vegetables 26

Stationery 19

Beauty and Cosmetics 21

Baby care 15

The process of gathering data and recording information is called data collection.

Annie wanted to represent the data using tally marks. Let us see!

Category

Tally Marks

Number of Products

Dairy

11

Fruits and vegetables

26

Stationery

19

Beauty and Cosmetics

21

Baby care

15 Important terms

Raw data

When the data is collected in its original form then it is called raw data.

Primary data

Primary data is the data collected by the user or researcher themselves.

Secondary data

Secondary data is the data collected by someone else. The main sources of secondary data are: 1

Research organisations

2

International organisations

3

Government bodies

Observation

Each item in the raw data is called an observation.

Array

When the raw data is arranged in ascending order or descending order it is called an array.

Frequency

The frequency is the number of times an observation occurs.

Frequency

Frequency distribution is the tabular arrangement of the numerical data which shows

distribution table Range

the frequency of each observation. The table showing the data is called the frequency distribution table.

The difference between the largest and the smallest observation in a set of data is called the range of the data.

Chapter 5 • Data Handling

65

Example 1

The scores obtained in 20 throws of a dice are: 5, 4, 3, 2, 1, 1, 2, 5, 4, 6, 6, 6, 3, 2, 1, 4, 3, 2, 2, 4 Organise the data and represent it in the form of a frequency distribution table.

First, arrange the data in ascending order, we get: 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6 The information can also be represented in tabular form as given below.

Scores

Example 2

Tally Marks

Frequency

1

3

2

5

3

3

4

4

5

2

6

3

The heights of 25 students (in cm) of class VII are as follows: 155, 158, 154, 162, 159, 151, 148, 156, 150, 152, 149, 160, 159, 163, 148, 147, 161, 155, 153, 156, 162, 159, 152, 159, 152 Prepare the frequency distribution table for the given data and answer the questions. a

What is the frequency of 159 cm?

b

What is the difference between the frequency of 153 cm and 162 cm?

c

Which observation has the highest frequency?

d

How many students have a height of more than 154 cm?

First, arrange the above data in ascending order. We get:

Heights

147, 148, 148, 149, 150, 151, 152, 152, 152, 153, 154, 155, 155, 156, 156, 158, 159, 159, 159, 159, 160, 161, 162, 162, 163

147

1

148

2

149

1

The information can also be represented in tabular form as shown.

150

1

151

1

152

3

153

1

154

1

155

2

156

2

158

1

159

4

160

1

161

1

162

2

163

1

a

The frequency of 159 cm is 4.

b

The frequency of 162 cm is 2 and that of 153 cm is 1.

So, the difference between the frequency of 153 cm and 162 cm is 2 – 1 = 1.

c

Height of 159 cm has the highest frequency.

d

14 students are more than 154 cm in height.

The maximum height of class VII student is 163 cm.

The minimum height of class VII student is 147 cm.

e

66

So, the range of the data is 163 cm – 147 cm = 16 cm

Tally Marks

Number of Students

Do It Together

A new movie was released last Friday. Ratings (out of 10) by 31 people who saw the movie are shown below: 3, 4, 4, 1, 2, 6, 3, 1, 4, 5, 1, 1, 2, 3, 2, 2, 6, 3, 2, 3, 2, 4, 4, 2, 5, 5, 2, 3, 2, 4, 4 a

Organise the data and represent it in the form of a frequency distribution table.

b

Find the range of the data.

Arranging the above data in ascending order, we get: 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6 The above information can also be represented in tabular form as given below.

Rating

Tally Marks

Frequency

1

4

2 3 4 5 6

Range of data = Highest rating – Lowest rating = 6 – 1 = 5

Do It Yourself 5A 1

2

Which one of of these steps is the first step in any investigation? a Organisation of data

b Collection of data

c Tabulation of data

d Understanding the data

A table which lists observations and uses tally marks to record and show the number of times they occur is called

_______________________________. a Frequency

3

b Tally marks table

c Frequency distribution table

Write (T) for true and (F) for false. Give reasons for the false statements. a Interviews, surveys and direct observations are examples of primary data. b Information published in magazines and newspapers after an event is an example of secondary data. c Autobiographies and memoirs are examples of primary data.

4

The data shows the colour of cushion covers chosen by 25 women from a local shop. Represent the data in a frequency distribution table.

Red, Blue, Red, Green, Yellow, Red, Blue, Blue, Green, Maroon, Black, Brown, Red, Brown, Black, Black, Yellow, Blue, Blue, Red, Brown, Yellow, Red, Green, Green 5

The data gives the number of birthday cakes sold by a bakery for 15 consecutive days. Prepare a frequency distribution table for the data. Find the range of the data given.

5, 4, 3, 2, 1, 1, 2, 5, 4, 6, 6, 6, 5, 2, 1

Chapter 5 • Data Handling

67

6

The marks obtained by 20 students in a mathematics test (out of 100) are given below: 45, 65, 63, 38, 46, 87, 72, 43, 64, 79, 91, 79, 76, 33, 82, 56, 69, 99, 82, 99 a Prepare a frequency distribution table and find the range of this data. b If 45 is the passing mark, how many have failed? c How many have scored less than 60 marks?

Word Problem 1

The students of a class were tested to find their pulse rate. The data were collected for the number of beats per minute.

60, 70, 70, 68, 67, 62, 71, 71, 59, 73, 73, 60, 60, 62, 62, 62, 68, 70, 70, 70, 74, 69. a Create a frequency distribution table for the data and find the range of the data. b How many students’ pulse rate was 60 or less? c How many students had a pulse rate of more than 65?

Making Bar Graphs Annie recorded the data of the stock available at her warehouse for different categories of products in a table.

Annie wondered how to represent the data using a bar graph.

Let us find out.

Category

Stock available

Dairy

1200

Fruit and vegetables

4600

Stationery

2600

Step 1

Step 3

Beauty and Cosmetics

1800

Draw the x-axis

Draw the rectangular bars.

Baby care

1600

and y-axis.

y

Scale: 1 division = 500

5000 4500 4000

Step 2 Label both the axes.

68

Stock Available

3500 3000 2500 2000 1500 1000 500 0

Dairy

Fruits and Stationery vegetables Category

Beauty and cosmetics

Baby Care

x

Double Bar Graphs Ria made a list of pizza toppings according to the choice of boys and girls.

Favorite pizza toppings

Girls

Boys

Cheese

15

12

Capsicum

13

15

Onions

18

20

Remember!

The double bar graph is drawn to compare two data sets.

Here, two sets of data are given. So, if we want to represent how many boys and how many girls choose the same type of topping, we need to draw two bars of different colours side by side for boys and girls separately to compare two sets of data simultaneously. This is known as a Double bar graph.

Let us learn how to draw a double bar graph for the two data sets given above: Step 1:

Step 2:

Draw two perpendicular axes – horizontal and vertical

Take the scale as 1 division = 5 children

Step 3:

Step 4:

Draw the bars of equal width with equal spacing between

Draw two bars for each type of topping, first for the girls

number of children.

on the graph.

and label both axes.

types of pizza toppings and heights corresponding to the

and second for the boys. Place them next to each other

y Scale: 1 division = 5 children

Number of Children

20

15 Girls 10

Boys

5

0

Cheese

Capsicum

Onions

x

Pizza Toppings

Chapter 5 • Data Handling

69

Error Alert!

The bars should be of the same width and there should be equal space between them. y 30

The first bar graph

25

25

appears in a book

20

20

15

15

by William Playfair

10

10

5

5

0

Example 3

Did You Know?

y

30

Pen

x

Pencil Book

0

in 1786.

Pen

Pencil

Book

x

The runs scored by a cricketer in the first six overs are given: Overs

I

II

III

IV

V

VI

Runs

10

16

12

8

10

5

Draw a bar graph for the data.

The runs a cricketer scored in the first six overs are shown in the bar graph below. y

Scale: 1 division = 5 runs

Number of Runs

20

15

10

5

0

I

II

III

IV

V

VI

x

Overs Example 4

Create a bar graph representing the number of orders received on a Monday by the food court inside a mall. The sales data are given in the table below.

Item No. of orders

70

Sandwich

Chow mein

Flavoured milk

Idli

300

500

200

550

From the table, we observe that the values of the sales data are too large to be represented.

A different scale would be required for this data set.

The bar graph below represents the number of orders received by the food court on a Monday. y

Scale: 1 division = 50 orders

600 550

Number of Dishes Ordered

500 450 400 350 300 250 200 150 100 50 0

Sandwhich

Chow mein

Flavoured milk

Idli

x

Item Example 5

The data shows the number of motorbikes of the same brand sold by two dealers in the first three months of a year.

Draw a double bar graph, choosing an appropriate scale.

Month

January

February

March

Dealer I

11

19

24

Dealer II

14

29

16

The double bar graph shows the number of motorbikes of the same brand sold by two dealers in the first three months of a year. y

Scale: 1 division = 5 motorbikes

Number of Motorbikes Sold

35 30 25 20

Dealer I Dealer II

15 10 5 0

January

February

March

x

Months

Chapter 5 • Data Handling

71

Example 6

Consider this data collected from a survey of a colony.

Favourite Sport

Cricket

Basketball

Swimming

Hockey

Athletics

Watching

1000

470

510

520

250

Participating

620

350

260

370

200

Draw a double bar graph choosing an appropriate scale.

This bar graph represents the number of people who are watching and participating in their favourite sports.

Let us consider, Scale: 1 division = 100 people y

Scale: 1 division = 100 people

1000 900

Number of People

800 700 600

Watching

500

Participating

400 300 200 100 0

Cricket

Basketball Swimming

Hockey

Athletics

x

Favourite Sport Do It Together

60 students from a certain locality use different modes of travel to school as given below: Mode of travel

Car

Bus

Scooter

Bicycle

Rickshaw

Number of students

6

15

10

18

11

Draw the bar graph representing the above data.

Mode of transport used by students y

Scale: 1 division = 2 Students

20

Number of Students

18 16 14 12 10 8 6 4 2 0

Car

Bus

Scooter

Bisycle

Mode of Transport

72

Rickshaw

x

Do It Yourself 5B 1

Fill in the blanks. a A bar graph consists of rectangular bars of equal _______________. b The space between two consecutive bars is _______________. c In a bar graph, the bars can be represented either _______________ or _______________. d The _______________ of a bar represents the frequency of the corresponding data.

2

3

The table shows the number of points scored by Rahul in 10 rounds of a game. Represent the data using a bar graph.

Game number

1

2

3

4

5

6

7

8

9

10

Points

6

18

2

14

10

8

10

14

16

19

The table shows the favourite sport of 300 students of a school. Represent the data using a bar graph.

Sport Number of Students

4

Football

Tennis

Athletics

Swimming

Hockey

Volleyball

80

60

25

30

15

40

50

The table shows the average intake of nutrients in calories by rural and urban groups in a particular year. Using a suitable scale for the given data, draw a double bar graph to compare the data.

Foodstuff

5

Cricket

Pulses

Leafy vegetables

Other vegetables

Fruit

Milk

Fish and flesh foods

Fats and oils

Sugar

Rural

35

14

51

35

70

10

9

19

Urban

49

21

27

30

50

37

17

13

The performance of a student in the 1st term and 2nd term is given. Draw a double bar graph choosing the appropriate scale.

Subject

Maths

Science

English

Social Science

Hindi

1st term

85

89

93

80

88

95

93

92

85

83

nd

2

term

Word Problem 1

2

Choose a suitable scale and draw a bar graph for the data below.

Day

Monday

Tuesday Wednesday Thursday

Friday

Number of toys manufactured

15,000

25,000

42,000

35,000

30,000

The data on the production of paper (in lakh tonnes) by two different companies X and Y over the years is given. Read the data and represent the data in the form of a double bar graph.

Years

2018

2019

2020

2021

2022

Company Y

25

35

30

40

55

Company X

Chapter 5 • Data Handling

30

40

25

30

45

73

Interpreting Bar Graphs Annie stored the data for the total revenue generated in the last year for the products in the form of a bar graph. y

Scale: 1 division = ₹50,000

500 000 450 000

Revenue Generated (in ₹)

400 000 350 000 300 000 250 000 200 000 150 000 100 000 50 000 0

Dairy

Fruits and Vegetables

Stationery

Beauty and Cosmetics

Baby Care

x

CATEGORY Annie wanted to check which category generated the most revenue. Let’s see.

The rectangular bar for fruit and vegetables is the highest.

The amount of revenue generated for fruit and vegetables is ₹4,00,000.

Horizontal bar graph Unlike vertical bar graphs or column graphs, there is another type of bar graph known as a horizontal bar graph.

A horizontal bar graph is drawn with rectangular bars of lengths proportional to the values that they represent, the same as a vertical bar graph.

The only difference is that, in the horizontal bar graph, the vertical axis shows the data categories that are being compared and the horizontal axis represents the values corresponding to each data category.

Think and Tell

Let us study an example to understand.

displayed in a bar graph?

The bar graph represents the data collected on the popularity of the types of music liked by students. Observing the bar graph, answer the given questions: 74

Which type of data can be

Is this a vertical or horizontal bar graph?

b

How many students like rock music?

Hip hop? Jazz? Classical?

c

Which type of music is the most popular?

d

How many students were asked about their favourite music?

It is clear from the horizontal bar graph that,

a

b

y

Scale: 1 division = 5 children

Hip hop Type of Music

a

Classical

Rock

Jazz

Here, the vertical axis displays the music category while the horizontal axis represents the corresponding value of each music category. So, this is a horizontal bar graph.

2

4

6

8

10

12

14

16

x

Number of Student

7 students like rock music, 12 students like Hip hop, 14 students like Jazz and 8 students like classical music.

c

Jazz is the most popular type of music among the students.

d

Total number of students = 12 + 8 + 7 + 14 = 41

Error Alert!

Do not forget to give the title and label the axes.

So, 41 students were asked about their favourite music.

The bar graph depicts the pass percentage of five different students in a class examination. Observing the bar graph, answer the questions: y Scale: 1 division = 10 % 100 90 80 70 Marks (in %)

Example 7

60 50 40 30 20 10 0

Suman

Raman

Prem

Ankita

Rosy

x

Student’s Name Chapter 5 • Data Handling

75

a

Name the student whose pass percentage is more than 90%.

b

Name the student who has the lowest pass percentage.

c

Find the number of students whose pass percentage is less than 80%.

d

Find the number of students whose pass percentage is more than 70%.

It is clear from the bar graph above that,

Example 8

a

Prem is the student whose pass percentage is 100% which is more than 90%.

b

Suman has the lowest pass percentage which is 50%.

c

Suman and Raman are the two such students whose pass percentage is less than 80%.

d

Raman, Prem, Ankita and Rosy are the students whose pass percentage is more than 70%. Thus, 4 students pass with more than 70%.

The double bar graph shows the favourite sports of boys and girls. Study the graph carefully and answer the questions.

Favourite Sports of Boys and Girls

y

Scale: 1 division = 2 students

16

Number of Students

14 12 10

Girls

8

Boys

6 4 2 0

Football

Tennis

Baskeball

Cricket

x

Sports a

In which sport do boys have the most interest?

b

Who has more interest in basketball?

Think and Tell

c

How many students in total like football?

Does the width of the bar in a bar

d

Which sports do 12 girls like?

e

How many girls like cricket?

It is clear from the bar graph given above that,

76

a

Boys have shown more interest in cricket.

b

Girls have shown more interest in basketball.

c

There are a total of 19 students who like football.

d

Basketball is liked by 12 girls.

e

Cricket is liked by 5 girls.

graph have any significance?

The bar graph shows the favourite fruit of the students in a class. Read the bar graph and answer the questions. y Scale: 1 division = 2 students a How many students like papaya? b

Which fruit is the favourite among students? How many students like it?

c

What is the total number of students?

d

How many more people like bananas than grapes?

e

What per cent of the total students like oranges?

f

What is the ratio of the number of students who like papayas to the number of students who like pears?

Oranges Pears Fruits

Example 9

Grapes Papaya Bananas Apples 2

4

6

8

10

12

14

x

Number of Students

a

Papaya is liked by 10 students.

b

The rectangular bar for bananas is the longest. So, most students like bananas. 12 students like bananas.

c

Students that like apples = 8

Students that like bananas = 12

Students that like papaya = 10

Students that like grapes = 7

Students that like pears = 5

Students that like oranges = 6

Total number of students = 8 + 12 + 10 + 7 + 5 + 6 = 48 d

People who like bananas = 12 People who like grapes = 7 The difference between the students who like bananas and those who like grapes = 12 – 7 = 5

e

Total number of students = 48 Number of students who like oranges = 6 Percentage = =

f

Total number of students who like oranges × 100% Total number of students 6 × 100% = 12.5% 48

The number of students who like papaya 10 2 = = The number of students who like pears 5 1 So, the required ratio is 2:1.

Chapter 5 • Data Handling

77

Do It Together

The double bar graph represents the data collected on the popularity of Indoor Games and Outdoor Games as participant sports for students in Grades 5 to 8.

Popularity Survey: Indoor Games vs Outdoor Games y

Scale: 1 division = 10%

100 90

Percentage of Students

80 70 60 Indoor Games

50

Outdoor Games

40 30 20 10 0

5th

6th

7th

8th

x

Grade a

Is there any grade in which more students prefer indoor games?

b

What percentage of students prefer indoor games in Grade 6?

c

What percentage of students prefer outdoor games in Grade 7?

d

What percentage of students prefer outdoor games in Grade 8?

e

60% of students of which grade prefer outdoor games?

It is clear from the double bar graph given above that,

a

Yes grade 7 _______________, more _______________ students prefer indoor games.

b

_______________ of students prefer indoor games in Grade 6.

c

_______________ of students prefer outdoor games in Grade 7.

d

_______________ of students prefer outdoor games in Grade 8.

e

60% of grade _______________ students prefer outdoor games.

Do It Yourself 5C 1

78

1 cm on a scale of a bar graph represents 5 ft. Find the actual height of the building if this is represented by a bar of 5 cm height in the bar graph.

2

The double bar graph, shows the performance of a student in the 1st and 2nd term.

y

Scale: 1 division = 10 marks

100 90 80

Marks

70 60

1st term

50

2nd term

40 30 20 10 0

Hindi

English

Maths

Science Drawings

x

Subjects

a In which subject is the performance of the student in the 2nd term poorer than in the 1st term? b In which subject did students perform the best in term 1? How many marks did he score? c In which subject(s) did students perform the best in term 2? How many marks did they score? d What was the total marks scored in term 1? e What is the difference between the total marks scored in term 1 and term 2?

3

The bar graph shows the heights of six mountain peaks in India. y Scale: 1 division = 1000 m 12 000 11 000

9 000 8 000 7 000

Aconcagua

2 000

Mount Everest

3 000

Nanda Devi

4 000

Kanchenjunga

5 000

Annapurna

6 000 Nanga Parbat

Height (in metres)

10 000

1 000 x

0 Mountain Peaks

Chapter 5 • Data Handling

79

a Arrange the mountain peaks according to their heights in descending order. b What is the highest peak, and what is its height? c What is the ratio between the height of the highest and lowest peak? d Which mountain peak is the second highest?

4

The table shows the cost price and profits earned by a merchant in 4 different types of merchandise P, Q, R and S. Merchandise

P

Q

R

S

Cost price

30

60

20

80

Profit

50

20

30

80

a Choose an appropriate scale and draw a double bar graph for the given data. b Which merchandise gives the most profit on selling?

5

The bar graph shows the monthly expenditure of a family (in ₹) in a year. Read the bar graph and answer the questions. y

Scale: 1 division = ₹200

2 600 2 400

Monthly Expenditure (in ₹)

2 200 2 000 1 800 1 600 1 400 1 200 1 000 800 400 200 x ec D

ov N

ct O

Se p

Au g

l Ju

n Ju

ay M

Ap r

M ar

Fe b

Ja

n

0 Months

a What was the family’s expenditure in March? b In which month/s did the family spend the most? How much did they spend? c What was the total expenditure of the family in the full year? d What is the ratio of the expenditure in the first six months and the expenditure in the last 6 months?

6

The data shows the rainfall on different days of the week. Study the data and answer the questions.

Days Rainfall (in mm)

80

Monday

Tuesday

Wednesday

Thursday

Friday

10

9

8

12

18

a Draw a bar graph of the data. b On which day the rainfall the highest? c What is the ratio of the rainfall on the first two days and the last two days?

7

The bar graph shows the number of people

y

with their shirt sizes. Read the bar graph and

Scale: 1 division = 100 people

answer the questions.

a How many people wear medium-sized

Extra-Large

b How many people wear extra-large sized shirts?

c What is the ratio of the people who wear

Shirt Size

shirts?

large-sized shirts to those who wear

Large Medium Small

small-sized shirts?

d How many people are there in total?

100

200

300

400

500

x

Number of People The bar graph shows the number of cars sold by two showrooms in seven months.

y

Scale: 1 division = 15 cars

120 105 Number of cars sold

8

90 75

Showroom A

60

Showroom B

45 30 15 0

Jan

Feb

Mar

April

May

June

July

x

Months a How many cars in total were sold by showroom A in the first three months? b How many cars in total were sold by showroom B in the last three months? c Which showroom sold more cars in the first 4 months altogether? d What is the ratio of the total cars sold by showroom A and showroom B?

Chapter 5 • Data Handling

81

Word Problems 1

The data on the number of visitors to a museum for five months is shown. Draw a horizontal bar graph to show the data.

Months

January

February

March

April

May

300

250

300

400

350

Number of visitors

a How many visitors were there in January? b How many more visitors were there in May than in February? c What percentage of the total number of visitors visited in April?

2

The double bar graph shows the percentage of profits earned by two companies A and B over the years. Read the graph and answer the questions.

Profit earned by company A and company B over the years y Scale: 1 division = 10%

Profit earned (in percentage)

60 50 40

Company A

30 Company B

20 10 0

2011

2012

2013

2014

2015

2016

x

Years

a If the income of company A in 2011 was ₹1,42,500, what was its expenditure in that year? b If the expenditure of Company A in 2010 was ₹70 lakhs and the income of Company

A in 2010 was equal to its expenditure in 2011, what was the total income (in lakhs) of

Company A in 2010 and 2011 together? (Hint: consider income as the selling price and expenditure as the cost price)

Points to Remember

82

•

The process of gathering required data from relevant sources is called data collection.

•

Primary data is the information collected directly from a source and secondary data is the information collected from an external source or agency.

•

The frequency of a particular data value is the number of times the value occurs in the data set.

•

A frequency table is a table which lists observations and uses tally marks to record and show the number of times they occur.

•

The range is the difference between the highest and the lowest values of the data. It is also called the spread of the data.

•

A bar graph is another way of representing numerical data using bars of uniform width drawn vertically or horizontally with equal spacing between them.

•

It is very important to choose the right scale for the data.

•

While choosing the scale, two factors should always be kept in mind, namely, the nature of the data and the purpose for which the measure is required.

•

A bar graph is a pictorial representation of data using bars (rectangles) of equal width and varying heights with uniform spacing between them.

•

A bar graph is used to compare the observations of one set of data whereas a double bar graph is used to compare two sets of data simultaneously.

Math Lab Objective: To compare the marks obtained in all the subjects by a student in the first and the second unit test in class VII by drawing a double bar graph using paper cutting and pasting.

Setting: Individual Materials Required: graph paper, origami sheets, pencil, eraser, sketch pens/glitter pens, glue and scissors.

Method:

1

Collect your marks for all the subjects in each term and tabulate the data.

2

Draw two perpendicular axes on the graph paper and label them to show what they represent.

3

Choose an appropriate scale and cut the strips of origami sheets of different lengths (or heights) according to the marks obtained in different subjects in each term, representing one term with one colour say blue and the other term with another colour say green.

4

Paste these strips of paper adjacent to each other representing a double bar graph.

5

Now, refer to the double bar graph and answer the following questions:

In which subject did you improve your performance the most?

Chapter 5 • Data Handling

83

Chapter Checkup 1

The marks obtained by 21 pupils of a class in a monthly test are given below: 37, 21, 43, 16, 25, 21, 28, 32, 45, 14, 43, 43, 21, 21, 16, 21, 28, 32, 45, 45, 14 How many students obtained more than 28 marks?

2

Strauss was rolling a die. He rolled it 40 times and noted the number appearing each time as shown below.

1

3

5

6

6

3

5

4

1

6

2

5

3

4

6

1

5

5

6

1

1

2

2

3

5

2

4

5

5

6

5

1

6

2

3

5

2

4

1

5

a Draw a frequency distribution table for the data. b Which digit appeared the least number of times? c Which digit appeared the most number of times? d Find the digits which appeared the same number of times.

3

Month

The table shows the number of children who have birthdays in four different months. Observe the data and answer the

Numbr of Childern

February

questions.

July

a Which month has the highest number of birthdays?

September

b What is the total number of birthdays in February and July?

November

c How many more birthdays fall in September than in

December?

December

d Which months have an equal number of birthdays?

4

5

84

The birth rate per thousand of five countries over some time is shown below. Represent the data with a horizontal bar graph.

Countries

India

Australia

Belgium

Canada

Denmark

Population (per thousands)

45

25

30

15

20

The data shows the total population of India based on the census conducted from 1971 to 2021, after every 10 years. Draw a bar graph for the given data.

Year

1971

1981

1991

2001

2011

2021

Population (in Millions)

420

540

680

1020

1200

1460

6

The bar graph shows the production of food grains in an Indian state for five consecutive years. Read the bar graph and answer the given questions.

Production of Food (in million tonnes)

y

Scale: 1 division = 10 million tonnes

100 90 80 70 60 50 40 30 20 10 0

2014

2015

2016

2017

2018

x

Years a In which year the production highest? What was the production in that year? b What was the production in 2016? c What was the difference between the production of 2016 and 2017? d What is the ratio of the production of 2015 to the total production?

The bar graph represents the number of people who use different electronic appliances. Study the graph and answer the given questions.

y

Scale: 1 division = 2 lakhs

12 Number of users (in lakhs)

7

10 8 6 4 2 0

Computer

TV

Telephone

iPod

Webcam

x

Type of electronic device a How many more people use radios than telephones? b How many more people use computers than iPods? c Find the difference between the number of users of iPods and webcams.

Chapter 5 • Data Handling

85

d Find the total number of people who participated in this survey. e Give the graph a suitable title.

8

The number of electric bulbs sold in a store over a week is depicted in the bar graph. y

Scale: 1 division = 25 Bulbs

250

Number of bulbs sold

225 200 175 150 125 100 75 50 25 0

Monday

Tuesday Wednesday Thursday

Friday

x

Days a How many bulbs were sold on Monday? b On which day was the highest number of bulbs sold and how many? c What fraction of the total bulbs were sold on Tuesday? d How many bulbs were sold in the entire week? e What is the average number of bulbs sold over the week?

9

In the bar graph, the salaries of 5 employees are shown. Spot the error and explain why this bar graph is misleading.

Represent the data in tabular form and redraw the correct graph.

Salaries (in thousands)

y

Scale: 1 division = ₹1000

28 27 26 25 24 23 22 21 20 19

Raj

Sahil

Nivira Employees

86

Yuvan

Kabir

x

10

Look at the bar graph which shows the approximate length of some national highways in India. Read the bar graph and answer the questions. y

Scale: 1 division = 200 km

N.H. 10

National Highways

N.H. 9

N.H. 8

N.H. 3

N.H. 2 x 0

200

400

600

800 1 000 1 200 1 400 1 600 1 800 2 000

Length (in km) a Which national highway is the shortest among the above? b What is the length of National Highway 9? c The length of which National Highway is about three times the length of National Highway 10?

Word Problems 1 The marks obtained by Kunal in his annual examination are shown below.

Subject

Maths

Science

Hindi

English

Social Science

Drawings

Marks

90

85

80

88

95

99

Draw a bar graph to represent the given data.

2 The bar graph given below shows the sales of books (in thousands) from six

branches of a publishing company during two consecutive years, 2010 and 2011.

Read the double bar graph and answer the questions that follow.

Chapter 5 • Data Handling

87

y Scale: 1 division = 20,000

Sales (in thousands)

140 120 100

2010

80

2011

60 40 20 0

B1

B2

B3

B4

B5

B6

x

Branches

a What information is given in the double bar graph? b What is the ratio of the total sales of branch B2 for both years to the total sales

of branch B4 for both years?

c What are the average sales of all the branches (in thousands) for the year 2010? d What are the total sales of branches B1, B3 and B5 together for both years

(in thousands)?

88

6

Introduction to Probability

Let’s Recall In a classroom of 14 students, there are 6 boys and 8 girls. We can find the ratio of the number of boys and girls and write in the simplest form.

a

Ratio of girls to boys =

Girls in the classroom 8 = Boys in the classroom 6

On reducing to the simplest form, we get b

Boys in the classroom Total number of students in the classroom 6 = 14

Ratio of boys in the classroom to the total number of students =

On reducing to the simplest form, we get c

8 8÷2 4 = = 6 6÷2 3

6 6÷2 3 = = 14 14 ÷ 2 7

Ratio of girls in the classroom to the total number of students = = On reducing to the simplest form, we get

8 8÷2 4 = = 14 14 ÷ 2 7

Girls in the classroom Total number of students in the classroom 8 14

Let’s Warm-up Match the ratios to their simplest forms.

Ratios

Simplest Form

1

6:12

1:5

2

5:25

4:5

3

3:9

1:2

4

8:10

1:4

4

4:16

1:3

I scored ____________ out of 5.

Understanding Probability Real Life Connect

The twins, Ajay and Rocky get a box of marbles for their birthday. They are distributing these marbles among themselves.

Ajay: Rocky, how do we distribute these marbles? I want all of the red marbles.

Rocky: Ajay! Let’s put our hand inside the bag and pick one marble at a time.

I’m sure you will get red every time you pick a marble.

Ajay: I think it’s impossible to pick a red marble each time, Rocky. I see very few red marbles in the bag.

We can infer from the above conversation that Ajay and Rocky are trying to predict whether Ajay will be able to pick all of the red marbles.

In various life situations, we can often not accurately predict the outcome, whether the event is sure to happen or impossible. In such cases, we use words like sure, impossible, likely, unlikely or equally likely.

These terms convey the uncertainty about the happening of an event. To measure this uncertainty, we use the concept of probability.

Chance Chance is the likelihood of some event happening. It can be expressed in words such as sure, impossible, unlikely, even chance, likely and certain.

Let us see some examples.

Impossible Event: Bird flying underwater

Sure Event: The sun rising in the East

When events have more, less or an even a chance of happening, they can be called likely, unlikely or equally likely, respectively.

Likely Event: Carrying an umbrella during monsoon

MONSOON SEASON

90

Unlikely Event: Child going to school on a Sunday

Equally likely Event: Tossing a coin and getting heads

The chance that an event will happen is called the probability of that event. Look at the spinner. Let’s look at the chances of the wheel stopping at different colours.

Wheel stopping at blue:

Wheel stopping at red:

Equally likely since 6 out of 12, which is half of the parts, is shaded in blue (light blue and dark blue).

Unlikely likely since 3 out of 12 parts are shaded in red.

Wheel stopping at green:

Wheel stopping at blue, red or yellow: Sure, since wherever the wheel stops, it will stop on one of these colours.

Impossible, since the colour is not on the wheel.

Example 1

Example 2

Do It Together

What is the chance of given events when you roll the dice? a

Getting a 3, 4 or 5. Equally likely

b

Getting a 6. Unlikely

c

Getting a number from 1 to 6. Sure

d

Getting the number 0. Impossible

Name one event for each of the given. a

Sure: Birds flying in the air

b

Impossible: A man flying in the air

c

Equally likely: Getting an odd or even number on rolling a dice

Describe the chance or likelihood of picking different colour balloons, without looking.

1

Equally likely Green balloon – _______________

2

White balloon – _______________

3

Green or orange balloon – _______________

4

Red balloon – _______________

Do It Yourself 6A 1

Write the chance of the given events happening using words like sure, impossible, likely, unlikely, and equally likely. a Getting a number greater than 5 on rolling a dice b Getting heads on tossing a coin c A lion flying in the sky d A child going to school on Monday e Picking a pencil from a box of pencils

Chapter 6 • Introduction to Probability

91

2

Ramesh spins the spinner as shown. What is the chance of the following events? a Getting an even number. b Getting the number from 1 to 10. c Getting the number 5 or 6. d Getting the number 20.

3

Ajay spins the spinner. What is the chance of the spinner stopping at: a yellow? b red? c green?

4

You are playing a game using the spinners shown. Answer the questions.

a You want to move down. On which spinner are you more likely to spin “Down”? Explain.

Spinner B

Forward

Forward

Forward

Up

Down

Reverse

Up

Down

b You want to move forward. Which spinner would you spin? Explain.

5

Spinner A

Down

Name one event for each of the given. a Likely

b Unlikely

c Impossible

Forward

Up Reverse

d Sure

Word Problem 1

Era draws a card from a set of number cards from 1 to 40. Write the chance of drawing the given set of number cards, in words. a Factor of 40.

b Even number card.

Theoretical Probability Ajay and Rocky want to know the probability of picking a red marble from the bag of marbles.

Let us help Ajay and Rocky understand how to find the probability of finding the red marble.

The probability of an event is the number that measures the chance or likelihood that an event will occur. Probabilities are between 0 and 1 including 0 and 1. We can sometimes measure probability using fractions.

Sure or Impossible Less likely Equally likely More likely Certain 0

92

1 4

1 2

3 4

1

Let us understand some terms Outcomes and Events Experiment: An action that produces clearly defined results is called an experiment. Flipping a coin or rolling a die is an example of an experiment.

Outcome: Each possible result of an experiment is called its outcome. For example, getting a tail when a coin is tossed is an outcome.

Event: A collection of one or more outcomes in an event.

Favourable Outcome: The outcome that is the same as what we guessed is called a favourable outcome.

Finding the probability of an event When all possible outcomes are equally likely, the probability of an event is the ratio of the number of favourable outcomes to the number of possible outcomes. The probability of an event is written as P(event).

P(event) =

number of favourable outcomes number of possible outcomes

This is also known as the theoretical probability.

Let us now find the probability of getting a red marble from the bag of 20 marbles. Given below are the marbles that were in the bag.

The bag has a total of 20 marbles. There are only 4 red marbles in the bag.

The chance of selecting a red marble is 4 out of 20. We can write the probability of the event as: Number of favourable outcomes Number of possible outcomes Example 3

4 1 = 20 5

Probability of an Event

Ramesh rolls the dice. What is the probability of rolling an odd number?

There are 6 numbers on a dice. 3 of them are odd numbers 1, 3 and 5. number of favourable outcomes number of possible outcomes

Did You Know?

3 6

theory of probability has its roots

P(event) = P (odd) =

1 On simplifying the fraction, we get, P (odd) = 2

The modern mathematical in attempts to analyse games of chance by Gerolamo Cardano in the sixteenth century.

Therefore, the probability of rolling an odd 3 1 number is or .

6 2 Chapter 6 • Introduction to Probability

93

Example 4

Rakesh randomly chooses one of the letters in the word EXPLORE. What is the theoretical probability of choosing a vowel?

P(event) =

number of favourable outcomes number of possible outcomes

There are 3 vowels in the word EXPLORE (e, o, and e).

There are a total of 7 letters.

3 7 There are 10 cards in a box which are marked with distinct numbers from 1 to 10. If a card is drawn without looking, what is the probability of drawing a prime number card?

Therefore, the probability of choosing a vowel is .

Do It Together

Cards in the box are 1, 2, _______________ The number of cards that have prime numbers are _______________ Total cards = _______________ Therefore, the probability of drawing a prime number card = _______________.

Do It Yourself 6B 1

Amit randomly chooses a letter from a hat that contains the letters A through K. a What are the possible outcomes? b What are the favourable outcomes of choosing a vowel?

2

Ajay spins the colour spinner. Answer the questions. a How many possible outcomes are there? b In how many ways can spinning red occur? c In how many ways can spinning not purple occur? What are the favourable outcomes of spinning not purple?

Sameer spins the number spinner shown. Answer the questions. a How many possible outcomes are there?

2

4

8

d In how many ways can spinning an odd number occur?

1

c In how many ways can spinning an even number occur?

5

3

b What are the favourable outcomes of spinning a number greater than 3?

7

3

6

e In how many ways can spinning a prime number occur?

d Spinning a multiple of 2

e Spinning a number less than 7

f Spinning a 9

6

c Spinning an odd number

1 5

b Spinning a 1

3

a Spinning red

2

94

Use the spinner to find the theoretical probability of the event for the given events.

4

4

5

A dodecahedron has twelve sides numbered 1 through 12. Find the probability and describe the likelihood of each event. a Rolling a 1

b Rolling a number less than 9

c Rolling a multiple of 3

d Rolling a number greater than 6

6

Each letter of the alphabet is printed on an index card. What is the theoretical probability of randomly choosing

7

There are 16 females and 20 males in a class. What is the theoretical probability that a randomly chosen student

8

A box contains 6 black crayons, 4 blue crayons, 5 red crayons, 3 yellow crayons, and 2 white crayons. One crayon

9

any letter except Z?

will be a female?

is chosen at random. Write each probability as a fraction. a P(black)

b P(blue)

c P(not white)

e P(black or blue)

f P(blue, red, or yellow)

d P(pink)

The numbers from 1 to 25 are written on slips of paper and one is selected at random. Write each probability as a fraction.

a P(odd)

b P(three-digit number)

e P(prime)

f P(greater than 19)

c P(not 4)

d P(positive)

10 What is the probability that a month picked at random ends with the letter y?

11 Amit calculates the probability of getting a number less than 3 when randomly choosing an integer from 1 to 10.

Favourable outcomes 3 = Total outcomes 10 Describe Amit’s error and give the correct probability. Write the numbers that are less than 3.

12 What is the probability of guessing the correct answer to a multiple-choice question if there are 5 choices?

13 What is the probability of guessing the correct answer to a true-false question?

Word Problems 1

Anand plays a word board game. He places 98 lettered tiles and 2 blank tiles in a bag.

Players will draw tiles from the bag one at a time without looking. What is the probability that the first tile drawn will be: a a blank tile?

2

b labelled with a letter?

There are 12 pieces of fruit in a bowl. Seven of the pieces are apples and two are peaches.

What is the probability that a randomly selected piece of fruit will not be an apple or a peach?

Chapter 6 • Introduction to Probability

95

Experimental Probability Ajay and Rocky think of checking how many red marbles Ajay gets when he picks marbles at random from the bag. They start picking the marbles one by one from the bag and list the frequencies in a table.

Marbles Picked

Frequency

Red

7

Blue

6

Green

5

Yellow

2

He is surprised to note that he picked red marbles as many as 7 times in 20 trials.

Probability that is based on such observations or repeated trials of an experiment is called experimental probability.

Think and Tell How does the experimental probability compare with the theoretical probability of picking a red marble?

Number of Times the Event Occurs P(event) = Total Number of Trials The table shows the results of spinning a coin 25 times. What is the experimental probability of spinning heads?

Heads

As the number of trials increases, the experimental probability gets closer to the theoretical probability.

The experimental probability of a red marble 7 being picked is .

20

Example 5

Remember!

Example 5

Tails

The table shows the results of rolling a dice 50 times. What is the experimental probability of rolling an odd number?

Number Rolled Frequency

6

6 25

96

8

11

11

6

So, an odd number was rolled 10 + 8 + 11 = 29 times in a total of 50 rolls.

P(event) = P(odd) =

Therefore, the experimental probability is Do It Together

4

The table shows 10 ones, 8 threes, and 11 fives.

19

Heads was spun 6 times in a total of 6 + 19 = 25 spins. number of times the event occurs P(event) = total number of trials P(heads) =

10

6 .

25

number of times the event occurs total number of trials

29 50

Therefore, the experimental probability is

The spinner shows 3 different colours and 3 numbers, one on each section. The spinner is spun 15 times and the outcome is recorded in the table below. Find the probability of the spinner stopping at the numbers 6 and 7.

Outcome

6

7

8

Number of Times

4

9

2

6

8 7

29 .

50

number of times the event occurs total number of trials P(6) = _______________ P(7) = _______________ P(event) =

The probability of the spinner stopping at 6 = _________. Probability of the spinner stopping at 7 = __________.

Do It Yourself 6C 1

Use the bar graph to find the experimental probability of the event. a Spinning a 6

10 Times Spun

b Spinning an even number c Spinning an odd number d Spinning a number less than 3 e Spinning a 7 f not spinning a 1 2

Spinning a Spinner

12 8 6 4 2 0

1

2

3 4 5 Number Spun

6

A spinner has four sections marked A, B, C, and D. The table shows the results of several spins. Find the experimental probability of spinning each letter as a fraction in its simplest form.

Letter

A

B

C

D

Frequency

14

7

11

8

3

Mala tossed a coin many times. She got 40 heads and 60 tails. She said the experimental probability of getting 40 heads was . Explain and correct her error.

60

4

Malati has a bag of marbles. She removes one marble at random, records the colour and then places it back in the bag. She repeats this process several times and records her results in the table. Find the experimental probability of drawing each colour.

Colour

Red

Blue

Green

Yellow

Frequency

12

10

15

13

5

There are 700 students at a high school. You survey 75 randomly selected students and find that 60 plan to go to

6

One hundred and twenty randomly selected students at a school are asked to name their favourite sport. The

college after completing high school. How many students are likely to go to college after completing high school?

results are shown in the table.

Sport

Cricket

Baseball

Football

Hockey

Other

30

22

34

20

14

Number of Responses

Find the experimental probability that a student selected at random makes the response given. a P(Cricket)

b P(hockey)

Chapter 6 • Introduction to Probability

c P(baseball)

d P(football)

97

In a survey, 125 people were asked to choose one card out of five cards labelled 1 to 5. The results are shown in

7

the table. Compare the theoretical probability and experimental probability of choosing a card with the number 1.

Cards Chosen Number

1

2

3

4

5

Frequency

15

30

35

20

25

The bar graph shows the results of spinning the spinner 200 times.

40

41

43

1

2 3 4 Number Spun

39

37

40

3

4

Times Spun

45

2

1

Spinning a Spinner 5

8

35 30 0

5

Compare the theoretical and experimental probabilities of the events. a Spinning a 4

b Spinning a 3

c Spinning a number greater than 4

d Spinning an odd number

Word Problem 1

A company produces electronic devices, and it is known that 5 out of 100 of the devices are defective. A quality control inspector randomly selects three devices from a batch of 100.

What is the probability that all three selected devices are defective?

Points to Remember

98

•

Probability measures the chance of an event happening.

•

An impossible event has no chance of happening.

•

An event with even chances has the same likelihood of happening and not happening.

•

A good chance event is more likely to happen than not to happen.

•

A certain event has a sure chance of happening.

•

An experiment is an investigation or a procedure that has varying results.

•

The possible results of an experiment are called outcomes.

•

A collection of one or more outcomes is an event.

•

The outcomes of a specific event are called favourable outcomes.

•

The probability of an event is a number that represents the likelihood that the event will occur.

•

Probability that is based on repeated trials of an experiment is called experimental probability.

Math Lab Rolling Dice!

Setting: In groups of 3.

Materials Required: one six-sided dice, paper, and pencil.

Method:

1

Write all possible outcomes of rolling a dice.

2

Roll the dice and note the outcome on paper.

3

Repeat this experiment 36 times.

4

Draw a table and list the frequency of outcomes in it.

5

Choose an expression from those in the list below that best describes the probability of each outcome.

Impossible

Unlikely

Even chance

Likely

Certain

We know that all numbers on a fair dice are equally probable. Has your dice turned out to be fair?

Chapter Checkup 1

Write the chance of the given events happening using words like sure or impossible.

2

What is the chance of the events happening? Fill in the blanks with words like sure, impossible, likely, unlikely or

a Kitten with wings

b Child wearing woollens in winter

equally likely.

a January comes before February is a/an _______________ event. b The sun setting in the East is a/an _______________ event. c A child going to school on a Monday is a/an _______________ event.

3

Use the labels on the right to describe the chance that a dice, when rolled, will show: a an odd number

b a6

c a number greater than 6

d a zero

e a number less than 10

Chapter 6 • Introduction to Probability

Impossible Unlikely

Equally likely

Likely

Sure

99

b A number less than 3 is not rolled

c An even number is rolled

d An even number is not rolled

e A 5 is not rolled

f A 7 is not rolled

You spin the spinner. Find the number of ways the event can occur and the favourable outcomes of the event. a Spinning a 1

b Spinning a 3

c Spinning an odd number

d Spinning an even number

e Spinning a number greater than 0

f Spinning a number less than 3

1

3

2

The bar graph shows the results of picking a numbered tile from

Picking a Numbered Tile

a bag 100 times. Use the bar graph to find the experimental

25 Times drawn

probability of the given events. a Picking a 2 b Picking an even number

20

20

22

21

19

3

4

5

18

15 10 5 0

c Not picking a 5

1

2

Number Picked

Use the spinner to find the theoretical probability of the events.

1

d Spinning a 4

8

c Spinning an even number

1

b Spinning a 1

4

a Spinning blue

1

10

2

6

d Picking a number less than 3 7

2

1

2

6

a A number less than 3 is rolled

3

5

A dice is rolled once. List the outcomes in each of these events.

3

4

Each student in your class voted for his or her favourite day of the week. Their votes are shown in the circle graph.

A student from your class is picked at random. What is the probability that this student’s favourite day of the week is Sunday?

Favourite Day of The Week Other, 6

Sunday, 6

8

9

Saturday, 10

Ten coloured discs are placed in a hat. Five are red, three are yellow, and two are black. If one disc is drawn from the hat, what is the probability that the disc will be: a Red?

b black?

e blue?

f red, yellow, or black?

c red or black?

d not black?

The 26 letters of the alphabet are written on cards and placed in a box. If one card is picked at random from the box, what is the chance that the letter on one of these letters is a X?

100

Friday, 8

b A vowel?

c M or N?

d A letter in the word MATHEMATICS?

10 The numbers from 1 to 25 are written on cards. If one card is chosen at random, what is the probability that the number on the card will be: a An odd number?

b A multiple of 5?

c A factor of 24?

d A number that contains the digit 2?

1 11 If the probability of an event is , how many times, on average, would you expect it to occur in 20 trials?

5 12 A bag contains five red balls numbered 1 to 5 and seven blue balls numbered 1 to 7. If a ball is chosen at random, what is the chance of choosing: a A red ball?

b A ball numbered 3?

d The blue ball numbered 1?

e An even-numbered ball?

c A ball numbered 6?

Word Problem 1 A library has 3000 fiction books and 4000 non-fiction books. What is the probability that a book selected at random is fiction?

N O N F I C T I O N

Chapter 6 • Introduction to Probability

F I C T I O N

101

37

Simple Equations

Let’s Recall We already know how to frame algebraic expressions.

Let us look at these pictures and try to guess the algebraic expressions formed by each of them.

We don’t know the amount of money in the bag. So let that amount be ₹x.

+

We also have a 10-rupee coin with us.

So, the total amount we have is ₹(x + 10). Here, x is the variable and 10 is a constant.

Now, we have two bags with the same amount of money in each. Let the amount of money in each bag be ₹x.

+

So, the sum of money in the two bags would be ₹2x.

We also have a 10-rupee coin.

So, the total amount of money we have is ₹(2x + 10).

Let’s Warm-up Match each statement with its expression.

Statement

Expression

3 more than twice a number

2n − 5

2 less than five times a number

3p + 2

2 more than thrice a number

2n + 3

5 less than twice a number

3m – 5

5 less than thrice a number

5n − 2

I Scored __________ out of 5.

Mean, and in Mode LinearMedian Equations One Variable Real Life Connect

Bhakti is a college student planning a summer vacation. To earn enough money for the trip, she has taken a part-time job at the local bank that pays ₹75 per hour. She opened a savings account with an initial deposit of ₹500 on January 1. If the summer vacation begins April 1, and the trip will cost approximately ₹11,000, how many hours will she have to work to earn enough to pay for her vacation?

Forming Linear Equations Let the number of hours worked by Bhakti be x.

She will be earning ₹75 per hour. So, the money earned by her would be ₹75x. She already has ₹500 in her savings account. So, the total money saved by Bhakti = 75x + 500. We know, cost of the trip = ₹11,000.

So, 75x + 500 = 11,000 This is an equation in x. You can get the minimum number of hours worked by Bhakti to save ₹11,000 by solving this equation. We will solve this equation later in this chapter using different methods.

Let us form a few more equations of this type. Consider this statement: ‘Seven added to a number equals 12’. We can write the statement in an equation as: x + 7 = 12.

In the expression (x + 7), the exponent of x is 1, so it is a linear expression.

Also, (x + 7) has only one variable. Hence, we can call x + 7 = 12 a linear equation in one variable. the variable

equality x + 7 = 12 x + 7 = LHS;

equation 12 = RHS

Remember!

The word linear means forming a line. The graph of a one variable equation with a highest power of one is a straight line. That is why such equations are called linear.

In an equation, the left-hand side (LHS) is equal to the right-hand side (RHS). Equations can have one or more variables, and each of those variables can have a power of one or more.

However, an equation which involves only one variable with the highest exponent as 1 is called a linear equation in one variable.

A linear equation in one variable is an equation that can be written in the form ax + b = 0, where a and b are constants and a ≠ 0.

For example: 2x + 3 = 7, x + 2 = 9, and a + 3 = 5.

What about x2 + 4 = 13? Is it a linear equation?

No, because variable x has exponent 2.

Chapter 7 • Simple Equations

Think and Tell

What kind of graph will a variable with exponent 2 make?

103

x + 3y = 5 is a linear equation in two variables.

2x + 3y + z = 25 is a linear equation in three variables.

Example 1

Write the given mathematical statements as linear equations. a The sum of x and 8 gives 12.

b Three more than 2 times of x gives 17.

Sum of x and 8 is x + 8.

Two times x is 2x. Three more than two times x is 2x + 3.

Equation is x + 8 = 12.

Example 2

Equation is 2x + 3 = 17.

Convert the given equations into statements. a x+3=8

b 3x + 5 = 26

The sum of x and 3 gives 8.

Example 3

Five more than three times x gives 26.

Akash scored 15 out of 41 points for the home team in a basketball game. His teammates scored x points in total. Write an equation to represent the given situation.

Points scored by Aakash + Points scored by his teammates = 41 15 + x = 41; Thus, 15 + x = 41 represents the given situation.

Do It Together

Sahil scored twice as many runs as Aakash. Together, their runs fell four short of a century. Form an equation to represent the situation.

Let the runs scored by Aakash be x.

Sahil scored twice as many runs as Aakash = __________ Together, their runs fell four short of 100 = 100 − 4 = 96 This situation −can be represented in the form of an equation as: x + _____ = __________.

On simplifying both the sides of the equation, we get, ____=____.

Do It Yourself 7A 1

Which of the given are equations? a 5x + 5 = 35

2

3

104

b 3a – 7 + 24

c

5x 3 18 + = 2 2 5

d 8x −

1 43 = 2 2

Which of the equations are linear equations? a x + 8 = 13

b x+9

e x2 + 5 = 14

f 9x + 7y + z = 24

c 5x + 6 = 8

d x + 8y = 6

Tick the linear equations in one variable and cross the others. a x + 9 = 14

b x + 12

e x2 + 7 = 29

f 6x + 8y + 3z = 38

c 4x + 5y = 8

d 7x + 3 = 10

4

State True or False. a 3p + 2 < 7 is not an equation.

__________________

b 2a + 6 – 11 is an equation.

__________________

c 2x2 + 3 = 19 is a linear equation.

__________________

d 8x + 3y = 9 is a linear equation in one variable.

__________________

8 = 4 denotes a numerical equation. e 2

5

6

Form equations for each of the given statements. a A number added to half of itself is 9.

b The sum of thrice a number and 5 is 17.

c Half of a number is 15.

d Half of the sum of 32 times of a number and 5 is 27.

e Two more than the number of frogs is 8.

f The number of students increased by 10 is 38.

Write the given equations in statement form. a y – 9 = 81

b 4q – 5 = 15

e 4x + 12 = 28 7

__________________

1 y+2=8 f 4

c 30z = 1650

Match each statement with its equation.

Statement

Equation

a One-tenth of a number gives 7.

10 – x = 7

b 10 subtracted from a number gives 7.

x + 27 = 77 1 x=7 10 x – 10 = 7

c A number subtracted from 10 gives 7. d The sum of a number and 27 is 77.

8

d 9x + 5 = 50

Write an equation for the given situations. a I have m candies. Lina has 3 less candies than me. We both have 18 candies altogether.

__________________

b Tina is c years old. Her 12-year-old sister is 3 years older than Tina.

__________________

c Rajat is twice as old as his brother Rakesh. The sum of their ages is 18 years.

__________________

d The length of a rectangle having perimeter 18 cm is 3 cm more than its breadth.

__________________

Word Problem 1

The length of a rectangular hall is 4 metres less than 3 times the breadth of the hall. What

2

A hotel has x rooms on each floor. There are 217 rooms on 7 floors. Write an equation to

is the length if the breadth is b metres? represent this situation.

Solving Equations: Same Operation on Both Sides Earlier in this chapter we learnt how to set up an equation for the given situation. Now, we will learn how to solve the equation formed.

Chapter 7 • Simple Equations

105

a If any number is added to one side of an equation, then the same number must be added to the other

side of the equation to maintain the equality between the LHS and RHS of the equation.

b If any number is subtracted from one side of an equation, then the same number must be subtracted

from the other side of the equation to maintain the equality of the equation.

c

If one side of an equation is multiplied by a number, then the other side must also be multiplied by the same number to maintain the equality between the LHS and RHS of the equation.

d If one side of an equation is divided by a number, then the other side must also be divided by the same

number to maintain the equality between the LHS and RHS of the equation.

In Bhakti’s situation, we wanted to know the number of hours she would work to save for her trip.

We had set up an equation as: 75x + 500 = 11,000 where x is the minimum number of hours.

Let’s solve this equation to get the value of the variable x.

First subtract 500 from both sides of equation.

75x + 500 – 500 = 11,000 – 500 On simplifying both the sides, we get, 75x = 10,500 Divide both the sides by 75. This will give us just x on the LHS.

Remember!

While solving a two-step equation like this, we always add or subtract the same number from both sides of the equation, then we perform multiplication or division on both sides of the equation.

75x 10,500 = ; So, x = 140 75 75 Therefore, Bhakti would need to work for a minimum of 140 hours to save enough money for her trip.

Example 4

Solve x + 8 = −5.

We shall subtract 8 from both sides of the equation. x + 8 – 8 = −5 – 8 On simplifying both the sides, we get, x = −13

Example 5

Solve 13a – 7 = 19.

We shall add 7 to both sides.

13a – 7 + 7 = 19 + 7 On simplifying both the sides, we get, 13a = 26 We shall now divide both the sides by 13.

13a 26 = ; So, a = 2.

13 13

Do It Together

Sahil has twice as much money as Jayant has. Together they have ₹240. How much money does Jayant have?

Let the money with Jayant be ₹x.

Money with Sahil = Twice as much as Jayant = _____ So, the required equation is (x + _____) = 240.

On simplifying the equation, we get, So, x = _____ Thus, Jayant has ₹_____.

106

Do It Yourself 7B 1

2

Give the first step you will use to separate the variable and then solve the equation. a

x–7=0

b

x+9=0

c

x−2=6

d

x+5=3

e

y − 8 = −4

f

y−1=1

g

y+5=5

h

y + 6 = −6

Solve the equation. a 4a = 60 e

3

b =9 3 z 6 f = 7 5

b

9y = 12

c =5 8 p 5 g = 6 18 c

d 5x = 28 h

30q = −5

d

30p = 60 7

Give the steps to solve the equation. a −x + 14 = 21

b 2x + 12 = 64

c 4z – 6 = 3

4

x 1 f =6 x−1=4 2 2 Eleven added to five times a number gives 61. Find the number.

5

What would you do to make equation 1 and equation 2 equivalent?

e 5+

Equation 1: x −

1 X = +5 3 6

Equation 2: 6x – 2 = x + 30

6

Aakash solved the equation 12 + x = 20 and got 32. Is the answer correct? If not, find the correct answer.

7

Describe and solve a real-world situation for each of the given equations. a x – 75 = 150

b 249 + x = 350

c 9x = 72

Word Problem 1

Tina saved ₹125 from her pocket money. How much more money does she need to buy a

2

Sameer has 20 stamps more than three times the number of stamps Rakesh has. If both

3

Two small jugs and one large jug can hold 8 cups of water. One large jug can hold 2 cups

jewellery set that costs ₹200?

together have 140 stamps, how many stamps does Rakesh have?

more than the small jug. Find how many cups of water each jug can hold.

Solving Equations: Transposition We know the equation, 75x + 500 = 11,000 where x is the minimum number of hours. Instead of subtracting 500 from both sides of the equation, we can simply move 500 from the LHS to the RHS by reversing the operation from addition to subtraction.

75x = 11,000 – 500 On simplifying the equation, we get, 75x = 10,500 Chapter 7 • Simple Equations

107

Instead of dividing both sides by 75, we can simply move 75 from the LHS to the RHS by reversing the operation from multiplication to division.

10,500 x= 75 On further simplification, we get, x = 140 Thus, Bhakti would have to work for a minimum of 140 hours to save enough money for the trip.

Did You Know?

The idea of using letters to represent variables was first suggested by French mathematician François Viète (1540–1603) in his work.

We got the same value for the variable despite using different methods.

The method used above for solving the equation is called the Transposition method. In this method, we move terms from one side of the equation to the other with the purpose of separating the variable on one side and the constants on the other side.

However, while moving the terms from one side of the equation to the other we should change the operator sign as: + to −

− to +

× to ÷

÷ to ×

Remember!

Transpose means to change the position or order of two things. In the transposition method we shift a term from one side to the other.

Example 6

Solve 5x – 7 = 13

Example 7

Transposing −7 from LHS to RHS as +7, we get, 5x = 13 + 7 On simplifying the RHS, we get, 5x = 20 On transposing 5 from LHS to RHS, the operation changes from multiplication to division.

20 5 So, x = 4.

x=

Do It Together

x+8 = 13 3 On transposing 3 from LHS to RHS, the operation changes from division to multiplication.

Solve

x + 8 = 13 × 3 On simplifying the RHS, we get, x + 8 = 39 On transposing 8 from LHS to RHS, the operation changes from addition to subtraction. x = 39 – 8 So, x = 31

Solve 0.8(2x − 3) – 0.5x = 6 On opening the bracket and simplifying the LHS, we get, 1.6x – 2.4 – 0.5x = 6 On transposing −2.4 from LHS to RHS, the operation changes from subtraction to addition.

1.6x – 0.5x = 6 + 2.4 On simplifying both sides of the equation, we get, _________________

108

Error Alert!

The number outside a bracket should be moved to the other side before opening the bracket.

On transposing 1.1 from LHS to RHS, the operation changes from multiplication to division.

_________________ On simplifying the RHS, we get, x = ___

Do It Yourself 7C 1

2

Solve each equation by the transposition method. a x + 4 = 10

b a + 9 = 13

d n + 2.6 = 4

e x–9=3

2 5 = 7 7 f x – 5.4 = 12.8

g −14 + a = 27

h p–

i

d −7m = −21 g

4

x – 6.5 = 15

Solve each of the equations. a 4y = 20

3

5 7 = 9 9

c x+

w

−9

= −5

b 90 = 15y

c −9a = 108

e

f

j =6 8 g h = 20 3.2

i

x = −11 3 2x = 30 4.5

Solve the given equations. a 6x + 14 = 16

b 7 + 5x – 3x = 21

c 1.2x + 4.8x = 18

d 6(5x − 2) + 2 = 30

e 7(x − 4) + 13 = 6

f 10y – 0.2(7 − y) = 19

What is the number? a I am thinking of a number. If 60 is subtracted from it, the result is 4.

__________________

b The sum of 13 and an unknown number is 46.

__________________

c If a certain number decreased by 21 is 81.

__________________

d If the product of 7 and a number is 56.

__________________

e A number when divided by 7 gives the quotient 5.

__________________

f If 10 is subtracted from thrice a number, the answer is 35.

__________________

g A number is multiplied by 3 and then 5 is added to it to get the answer as 20.

__________________

5

Nine added to thrice a whole number gives 45. Find the number.

6

Two-thirds of a number is greater than one-third of the number by 3. Find the number.

7

Divide 38 into two parts so that one part is 18 more than the other.

8

The length of a rectangle is 5 m greater than its breadth, and its perimeter is 250 m. Find the length and breadth of the rectangle.

Chapter 7 • Simple Equations

109

Word Problem 1 2

A bat and a ball together cost ₹1200. The cost of the ball is

costs of the bat and the ball.

1 of the cost of the bat. Find the 5

A vegetable seller has 31 kg of onions. He divides the onions in two bags so that one bag weighs 5 kg more than the other. Find the weight of the bags.

Applications of Simple Equations In general, we come across many real-life problems which can be solved using the concept of linear equations in one variable.

Let us solve a real-world problem using some steps.

Kajal takes 10 rounds of a square garden. Find the side of the square if the total distance covered by her is 2560 m.

Step 1: Note down what is given and what is required.

Step 2: Denote the unknown by some variable.

Given: Garden is square.

Let the measure of each side of this square garden be x.

Required: Measure of each side of square garden.

Step 3: Translate the statements into algebraic expressions.

Taking 1 round of the square garden = perimeter of the garden = 4x

Step 4: Form an equation as per the condition given in the problem.

10 × 4x = 2560

Step 5: Solve the equation to find the value of the variable.

2560 ; x = 64 m 10 So, each side of the square garden measures 64 metres.

4x =

Example 8

The sum of two numbers is 25. If one of the numbers exceeds the other by 7, then find the numbers.

Let the first number be x; The second number will be x + 7.

As per the given condition, x + x + 7 = 25 On simplifying, we get, 2x + 7 = 25

x = 18; x = 9 2 so, one of the numbers is 9. The other number = x + 7 = 9 + 7 = 16.

So, the required numbers are 9 and 16.

Example 9

The sum of two consecutive numbers is 13. Find the numbers.

Let one number be x; Its consecutive number will be (x + 1).

As per the given condition, x + (x + 1) = 13 On simplifying, we get, 2x + 1 = 13

x = 12; x = 6 2 So, one of the numbers is 6. The other number is 6 + 1 = 7.

110

Example 10

The cost of three pens and four pencils is ₹248. If one pen costs ₹15 more than one pencil, then find the cost of each pen and each pencil.

A pen costs ₹15 more than a pencil. So, the cost of the pen is dependent on the cost of the pencil.

Hence, let us assume the cost of one pencil to be x.

Then the cost of a pen = 15 + x The cost of 4 pencils = 4x and the cost of 3 pens = 3(15 + x) Total cost of 3 pens and 4 pencils = ₹248 3(15 + x) + 4x = 248 On opening the brackets, we get, 45 + 3x + 4x = 248 On simplifying, we get, 45 + 7x = 248

x = 203 ; x = 29 7 Thus, the cost of a pencil is ₹29. And the cost of a pen = 15 + x = 15 + 29 = ₹44.

The cost of each pen is ₹44, and the cost of each pencil is ₹29.

Example 11

There are 50 paise and 20 paise coins in a purse. The number of 20 paise coins is 4 times that of 50 paise coins. If the total value of the money is ₹26, find the number of coins of each denomination.

Let the number of 50 paise coins be x.

Then, the number of 20 paise coins will be 4x.

Therefore, the value of x coins of 50 paise = 50x paise Also, the value of 4x coins of 20 paise = 20 × 4x paise = 80x paise The total value of money = ₹26 or 2600 paise As per the given condition, we have, 50x + 80x = 2600 On simplifying, we get,

x = 2600; x = 20 130 Since, x = 20, 4x = 4 × 20 = 80 Hence, there are 20 coins of 50 paise and 80 coins of 20 paise.

Do It Together

The length of a rectangular plot exceeds its breadth by 5 metres. If the perimeter of the plot is 142 metres, find the dimensions of the plot.

Let the breadth of the plot be x metres.

So, the length will be (x + 5) metres.

We know that perimeter of a rectangle = 2(Length + Breadth) As per the given condition, 2(x + 5 + x) = 142 metres

Chapter 7 • Simple Equations

111

On simplifying the LHS, we get, 2(2x + 5) = 142 OR __________ On transposing 10 from LHS to RHS, the operation changes from addition to subtraction.

4x = 142 − 10

On simplifying, we get, x = 132; x = _____ 4

Breadth of the rectangular plot is ___ metres.

Length of the rectangular plot is ___ metres.

Do It Yourself 7D 1

Twice the number decreased by 22 is 48. Find the number.

2

A number when added to its half gives 72. Find the number.

3

The sum of two numbers is 95. If one exceeds the other by 3, find the numbers.

4

Divide 20 into two parts so that one part is 8 more than the other. What are the two parts?

5

Find three consecutive numbers so that the sum of the second and the third exceeds the first by 24.

6

The angles of a triangle are x°, (x + 40)°, and (x − 10)°. Find the angles.

7

Three sides of a triangle measure (2x + 3) units, (x + 5) units and 4x units. Find the measure of the sides if its perimeter is 148 units.

Word Problem

112

1

The age of Olivia’s mother is 5 years more than three times Olivia’s age. Find Olivia’s age if

2

1 Five years from now, Satish will be 1 times his present age. How old is he now?

2

3

There are 320 books and magazines altogether in a bookshop. There are three times as

4

The cost of five books and three pens is ₹730. If a books costs ₹50 more than a pen, then

5

Heena’s mother is 4 times as old as she is. After 20 years, her age will be twice that of

6

Savita is Tina’s elder sister. The sum of the ages of Savita and Tina is 18 years and the

her mother is 44 years old.

many books as magazines. How many books are there?

find the cost of the book and the pen.

Heena’s. Find their present ages.

difference between their ages is 4 years. Find the ages of Savita and Tina.

Points to Remember •

An equation is a statement of equality which has one or more than one variable.

•

An equation in one variable is called a linear equation if the highest power of the variable is one.

•

The value of the variable of an equation is called the solution of the equation.

•

For solving an equation, we can a. add the same number on both sides. b. subtract the same number from both sides. c.

multiply both sides by the same non-zero number.

d. divide both sides by the same non-zero number.

•

The process of shifting a number from one side of the equation to the other by changing the sign is called transposition.

Math Lab Modelling Simple Equations!

Setting: In groups of 3.

Materials Required: Rectangular paper strips of blue, square paper strips of yellow and pink Method:

1

Make rectangular paper strips of blue and square paper strips of yellow and pink out of chart paper.

2

A blue rectangular strip represents a variable. A yellow square strip represents a positive constant and a pink square strip represents a negative constant.

3

A yellow and pink strip collectively represent zero.

4

Think of a linear equation, say 3x + 4 = 2x – 5.

5

Represent the linear equation using rectangular and square strips.

6

Subtract or remove strips to solve the equation.

7

Repeat the same process to find the solution for other linear equations.

=

Chapter Checkup 1

Find the value of the unknown.

3 3 a x+ =1 8 8 5 3 d x+ = 19 19

Chapter 7 • Simple Equations

b x−

1 2 = 3 3

e x – 3 = −20

c x+5=8 f

x–3=7

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2

Complete the equations by performing the same operations on both sides. z z a b x + 1 = −6. So, x = _____ − 2 = 26. So, = ______ 2 2 x+1 x+1 c d 2x + 11 = 17. So, 2x = _____ − 3 = 27. So, = ______ 5 5 e 5(x − 3) = 25. So, (x − 3) = _____ f 3(y − 7) – 4 = 14. So, 3(y − 7) = _____

3

Determine by substitution.

4

5

a −2 is the root of 5x = 10

b −5 is the root of x – 5 = 0

c 3 is the root of y + 22 = 25

d 1 is the root of 2q + 4 = 22

e −3 is the root of 3p + 3 = 13

f

g 2 is the root of 18x – 42 = 13

h 4 is the root of 6x – 8 = 16

9 is the root of 4a – 8 = 28

Solve the equations. a 9x = 18

b 4a = 12

c −3y = −27

d −18 = −6x

e −4x = 8

f

−7x = 3

Frame the equations and solve. a One and a half times a number is 150. What is the number? b I am thinking of a number. Five times that number is 65. What is that number? c 3 subtracted from 5 times of p gives 22. What is the value of p? d Double of a number added to 36 gives 48. What is the number? e 31 is 9 more than 4 times the number p. What is the value of p?

6

7

Solve. q a =7 6 e x + 10 = −10

x =2 −29 f x – 17 = −20

b

9 x = 54 2 g y+7=4

−13 x = 26 2 h x + 20 = −20

c

d

Find the root of each of the equations. x a b 3x – 2 = 5x – 12 −5=6 3 d −22 = 14 – 9x e 13y = −12y + 100

c 12y = 7y – 15 f

5(x − 3) = 10

g 9(x + 14) = 27

i

5x + 12 = 2

3−x h =6 5

8

When you divide a certain number by 13, the quotient is −18 and the remainder is 7. Find the number.

9

Find three consecutive even numbers whose is 96.

10 Find three consecutive numbers so that the sum of the second and the third number exceeds the first by 14.

11 If one side of a square is represented by 4x – 7 and the adjacent side is represented by 3x + 5, find the value of x.

12 Each of the two equal sides of an isosceles triangle is three times as large as the third side. If the perimeter of the triangle is 28 cm, find each side of the triangle.

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Word Problems 1

There are three packets of goods p, q, and r. The weight of packet p is 11 kg more 2 than that of packet q and the weight of packet r is 21 kg more than that of packet q.

2 If their total weight is 32 kg, find the weight of the packets.

2

The price of 2 notebooks and 3 textbooks is ₹120, but a notebook cost ₹10 more than the textbook. Find the cost of both the notebook and the textbook.

3

Ramesh has a certain number of toffees. His sister has double the number of toffees compared to Ramesh. If together they have 54 toffees, then how many toffees does Ramesh have?

4

A father is 30 years older than his son. In 12 years, the man will be three times as old as his son. Find their present ages.

Chapter 7 • Simple Equations

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