G8_DinoLab_Ebook

Acknowledgements Academic Authors: Muskan Panjwani, Anna Tsoy, Nadezhda Sysonova Creative Directors: Alena Sizintseva

Book Production: Anastasia Shnip, Anastasia Voitovich All products and brand names used in this book are trademarks, registered trademarks or trade names of their respective owners. © Uolo EdTech Private Limited First edition 2026 This book is sold subject to the condition that it shall not by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher’s prior written consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser and without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of both the copyright owner and the abovementioned publisher of this book.

Book Title: DinoLab Math Smartbook 8 ISBN: 978-93-89789-46-1 Published by: Uolo EdTech Private Limited Corporate Office Address: 91Springboard, 3rd Floor 145, Sector 44, Gurugram, Haryana 122003 CIN: U74999DL2020PTC360472 Printed by: Printpro Solutions All suggested use of the internet should be under adult supervision.

How to get access to DinoLab Get access to animated interactive courses, Marathons, Olympiads and much more — all in the Uolo Learn app!

1. Download the "Uolo Learn" app from Google Play (Android) or App Store (iPhone).

2. In the app, click scanner to scan the QR code below.

3. Follow the instructions in the app to access the content.

Class: Name: School:

Welcome to DinoLab! DinoLab is an AI-powered self-learning platform that helps children learn Mathematics and other subjects step by step, at their own pace. Students can practise and revise every topic digitally and through printed smartbooks.

The DinoLab Mathematics Smartbook is a companion to the digital course. Each exercise has a QR code linking to the Uolo Learn app for continued practice.

Using AI, DinoLab creates a personalised learning path: it explains the concept, gives guided practice and adapts if mistakes occur — helping students gain clear understanding.

Digital content is presented as interactive flashcards with 50,000+ gamified exercises and animations, making learning engaging and enjoyable.

DinoLab works in Uolo apps and on multiple devices

SMART

Uolo Mobile App

Smartboard

Computer Lab

Tablet and Laptop

Smartbooks

To use DinoLab on the web, Smartboards and in computer labs, your school will receive special access for each student and teacher.

ii

How to Use the DinoLab Solution Once the Uolo Learn app is installed and you are logged in, you can access DinoLab. Our Mathematics course is designed with interactive exercises that help children cover the school syllabus step by step, at their own pace.

1

Open the Uolo Learn app.

2

Tap on DinoLab.

3

Select Mathematics and start solving!

Compete and win in Marathons!

• • •

Solve problems and earn points Check the leaderboards of your class, school and all of India Get achievement certificates

QR codes in the smartbook

In this Smartbook, you will find QR codes placed next to the exercises.

Simply use the QR scanner inside the app to access the interactive content.

1

Open the Uolo Learn app.

2

Click on QR icon in the upper right corner.

3

Scan QR code next to the task.

n Sca

QR

e

cod

Curricular Programs

iii

CONTENTS

1. Rational Numbers Operations on Rational Numbers ���������������������������������������������������������������������� 1 2. Solving Equations in One Variable Solving Linear Equations ��������������������������������������������������������������������������������� 17 3. Polygons and Quadrilaterals Understanding Polygons and Quadrilaterals ������������������������������������������������ 27 4. Bar Graphs and Histograms Frequency Distribution of Grouped and Ungrouped Data ������������������������� 33 Graphical Representation of Ungrouped Data �������������������������������������������� 39 Graphical Representation of Grouped Data �������������������������������������������������� 47 5. Pie Charts Drawing and Reading Pie Charts �������������������������������������������������������������������� 53 6. Probability Understanding Probability ������������������������������������������������������������������������������ 61 7. Squares and Square Roots Square of a Number ����������������������������������������������������������������������������������������� 65 Square Roots ����������������������������������������������������������������������������������������������������� 69 8. Cubes and Cube Roots Cube of a Number �������������������������������������������������������������������������������������������� 73 Cube Root of a Number ����������������������������������������������������������������������������������� 79 9. Percentage and Its Applications Percentage and Percentage Change �������������������������������������������������������������� 89

iv

10. Compound Interest Simple Interest and Compound Interest ������������������������������������������������������101 11. Algebraic Expressions Addition and Subtraction of Algebraic Expressions �������������������������������������107 Multiplication of Algebraic Expressions ���������������������������������������������������������113 12. Area of Polygons Area of Figures Made with Polygons ��������������������������������������������������������������119 13. Surface Area and Volume of Solids Surface Area ����������������������������������������������������������������������������������������������������� 125 14. Exponents and Powers Exponents ��������������������������������������������������������������������������������������������������������� 129 15. Direct and Inverse Proportions Proportion ��������������������������������������������������������������������������������������������������������� 137 16. Factorisation and Division of Algebraic Expressions Factorisation of Algebraic Expressions ��������������������������������������������������������� 145 17. Linear Graphs Line Graphs and Linear Graphs ��������������������������������������������������������������������� 153 Answers ������������������������������������������������������������������������������������������������������������ 163

v

1. RATIONAL NUMBERS

OPERATIONS ON RATIONAL NUMBERS 1. Look at the shaded parts in each image. Fill in the blanks.

+

=

+

=

+

=

–

=

=

2. Complete the rule for adding rational numbers having a common denominator. Write the letters a, b, and c in the correct boxes.

3. Complete the rule for subtracting rational numbers having a common denominator. Fill in the blanks.

1

Rational Numbers

4. Look at the denominators of the rational numbers in each expression. Join each expression to the correct category.

Equal denominators

Different denominators

5. Simplify the expressions.

Rational Numbers

2

6. Compare the rational numbers in each pair.

Follow the steps and fill in the blanks.

Step 1. Find common denominator and write in the blank.

Step 1. Find common denominator and write in the blank.

Step 2. Convert the rational numbers to equivalent forms having a common denominator.

Step 2. Convert the rational numbers to equivalent forms having a common denominator.

=

=

Step 3. Compare the rational numbers you got. Put the same sign into the original expression.

=

=

Step 3. Compare the rational numbers you got. Put the same sign into the original expression.

7. Mark each rational number on the number line given below.

0

3

Rational Numbers

1

8. Compare the rational numbers in each pair. Follow the steps and fill in the blanks.

Step 1. Find common denominator and write in the blank.

Step 1. Find common denominator and write in the blank.

Step 2. Convert the rational numbers to equivalent forms having a common denominator.

Step 2. Convert the rational numbers to equivalent forms having a common denominator.

=

=

Step 3. Compare the rational numbers you got. Put the same sign into the original expression.

=

=

Step 3. Compare the rational numbers you got. Put the same sign into the original expression.

9. Mark each rational number on the number line given below.

0

1

Rational Numbers

4

10. Read the conversations. Help Sonu answer Eva's questions. Tick the correct option.

How do you find a common denominator for

and

?

To find a common denominator for fractions, you need to find a number that ……………………………………… a) is divisible by 3 as well as 5.

b) divides 3 as well as 5.

c) is the sum of 3 and 5.

d) is divisible by 1 as well as 2.

How do you find a common denominator for fractions

and

?

To find a common denominator for fractions, you need to find a number that ……………………………………… a) divides 9 as well as 4.

b) is the sum of 9 and 4.

c) is divisible by 9 as well as 4.

d) is divisible by 1 as well as 2.

How do you find a common denominator for fractions

and

?

To find a common denominator for fractions, you need to find a number that ……………………………………… a) divides b as well as d.

b) is divisible by a as well as c.

c) is the sum of b and d.

d) is divisible by b as well as d.

11. Look at the rational numbers in each set. Convert them to equivalent forms having the least common denominator.

5

Rational Numbers

12. Trace your path through the maze. Walk through only irreducible rational numbers.

Now reduce all the reducible rational numbers in the maze.

Write all the rational numbers that are greater than one.

Rational Numbers

6

13. Some rational expressions are simplified below.

Identify the mistake in each. Write the correct solution.

7

Rational Numbers

14. Simplify each expression on the puzzle pieces on the left. Then join it to the matching green piece on the right.

Rational Numbers

8

15. Calculate the area of the shaded region. Fill in the blanks.

1 How many parts are shaded?

1 How many parts are there in total?

16. Look at the partitions in the given shape.

Find the unknown side and the area of the shaded region.

Fill in the blanks.

1 The area of the shaded region:

1

9

Rational Numbers

17. Multiply. Write the product as a rational expression.

Fill in the blanks.

18. Join each product on the left to the correct answer on the right.

Rational Numbers

10

19. Multiply. Write the product in simplified form.

20. For each expression, which values are NOT allowed for the variable? Tick the correct option.

All the values of y are possible.

All the values of x are possible.

11

Rational Numbers

21. Use the data given in the table. Find the distance travelled by each car. Complete the table.

Purple

Speed (S)

Time (T)

Distance (D) = Speed (S) × Time (T)

Blue Red

Which car travels the farthest? Write your answer in the blank.

Rational Numbers

12

22. At a party, 4 pizzas were eaten. Each guest ate 2/3 of a pizza. How many guests were there at the party?

Answer

At a party, 3 pizzas were ordered. Each guest ate 2/3 of a pizza.

How many guests were there at the party? How much pizza was left?

Answer

23. Complete the rule to divide two rational numbers.

Fill in the blanks.

13

Rational Numbers

a

c

×

b

d

×

24. Are the following division statements correct? If not, underline the mistake in each.

25. Divide.

Rational Numbers

14

26. Change each division sentence to an equivalent multiplication sentence. Fill in the blanks.

27. Join each expression on the left to its equivalent expression on the right.

28. In an activity, children are pouring water into a rubber pool. In 60 seconds, they fill 1/30 of the pool.

What part of the pool will be filled in the given time?

Tick the correct option.

15

3 min

The pool will be completely filled.

The pool will overflow.

15 min

The pool will be completely filled.

The pool will overflow.

30 min

The pool will be completely filled.

The pool will overflow.

Rational Numbers

29. Simplify.

30. Simplify the expressions. Arrange the results on the ladder from smallest (at bottom) to largest (at top).

Rational Numbers

16

2. SOLVING EQUATIONS IN ONE VARIABLE SOLVING LINEAR EQUATIONS 1. Read each scenario. Write the correct expressions in each box.

Scenario 1

A game console costs x rupees and a game costs y rupees.

One game console and 2 games cost

rupees.

rupees more than a game.

One game console costs Scenario 2

The table shows the populations on 3 planets. The population on planet Alpha is the sum of the populations on planets Omega and Gamma.

Planets

Population

Alpha

7x

Omega

x

Gamma

(3x + 70)

Write an equation for the population on planet Alpha. +

=

Scenario 3

The table shows the populations on 2 planets. The population on planet Delta is one-third of the population on planet Zeta.

17

Planets

Population

Delta

y

Zeta

y + 350

Solving Equations in One Variable

Write an equation for the population on planet Delta. ×

=

2. Look at the equation on each piece of luggage. How many variables are in each? Join each piece of luggage to the correct category.

1 variable

more than 1 variable

3. Look at each equation in variable x. Join it to the castle that shows its correct form. Note that a, b are some constant numbers.

ax = b

ax + b = 0

other

Solving Equations in One Variable

18

4. Are the following statements true or false?

Tick the correct option.

a) If

Then

True

False

b) If

Then

True

False

c) If

Then

True

False

d) If

Then

True

False

5. Find the root of each equation given below. Tick the correct option.

A root of an equation is a value of the variable that makes the equation true.

When this value is substituted into the equation, both sides become equal.

6. Look at each of the given roots. Tick all the equations that have this root.

Root = 8

19

Solving Equations in One Variable

Root = 3

7. Look at each equation. Answer the given questions.

Tick the correct option. Then solve for x.

Which of these will isolate x?

Divide both sides by a.

Divide both sides by (x + a).

Subtract a from both sides.

Subtract x from both sides.

a)

b) Step 1: Which of these will transform –5x + 2 into –5x?

Subtract 2 from both sides.

Divide both sides by (–5).

Divide both sides by (–2).

Add 2 to both sides.

Step 2: Which of these will transform –5x into x?

Divide both sides by 5.

Multiply both sides by 5.

Divide both sides by (–5).

Multiply both sides by (–5).

c) Step 1:

Which of these will transform + 4 into

?

Divide both sides by

Multiply both sides by

.

Multiply both sides by 4.

Divide both sides by 4.

.

Step 2:

Which of these will transform into ?

Subtract 4 from both sides.

Add 8 to both sides.

Subtract 8 from both sides.

Divide both sides by .

8. Solve for x.

Solving Equations in One Variable

20

9. How many roots does each equation have? Tick the correct option.

no roots

two roots

no roots

two roots

one root

infinite roots

one root

infinite roots

no roots

two roots

no roots

two roots

one root

infinite roots

one root

infinite roots

10. Look at each balance. What value of x keeps it balanced? Write your answers in the blanks.

x

1 x x

1 1 1 1 1 1 1 1 1 1

x x

1 1 1

1 1 1 1 1

11. Write an equation of the form ax = b using the given values of a and b. Then solve it.

21

Solving Equations in One Variable

12. Complete the flow chart to solve the equations of the form ax + b = 0. Join each empty box to the correct step.

a = 0? yes

no

b = 0? no

x — any number

yes

no roots

13. Solve the equations.

Solving Equations in One Variable

22

14. Solve for x.

15. Solve for x. Write your answers as decimal numbers.

16. Solve.

23

Solving Equations in One Variable

17. The first chapter of an audiobook is 16 minutes longer than the second chapter. The third chapter is twice as long as the second chapter. Listening to all three chapters takes 40 minutes. How many minutes does the second chapter last?

18. At a Halloween party, three friends Sam, Eva and Myra collected candies. Sam collected three times as many candies as Myra. Eva collected 20 more candies than Myra. Sam and Eva collected an equal number of candies.

How many candies did Myra collect?

19. Find the value of x using the cross-multiplication method.

Solving Equations in One Variable

24

20. Solve the equations.

25

Solving Equations in One Variable

Solving Equations in One Variable

26

3. POLYGONS AND QUADRILATERALS

DATA HANDLING

UNDERSTANDING POLYGONS AND QUADRILATERALS 1. Tick the correct answers.

What condition must be met by a closed figure to be a polygon?

It must be made up of line segments only.

The number of sides must be five or more.

Three sides must come out of one vertex.

The sides must not cross each other.

Draw 2 polygons here.

Draw 2 shapes that are not polygons.

2. Find the figures that can be completed to form a quadrilateral. Add the missing sides to turn them into quadrilaterals.

27

Polygons and Quadrilaterals

3. Match each quadrilateral with its type. Write your answer in the blank.

trapezium

parallelogram

quadrilateral

rhombus

rectangle

square

4. Join each quadrilateral on the left to its description on the right.

trapezium

at least one pair of parallel sides

quadrilateral

any 4 sided polygon

rectangle

2 pairs of parallel sides

parallelogram

equal angles, 2 pairs of parallel and equal sides

rhombus

equal sides and equal angles

square

parallelogram with equal sides

Polygons and Quadrilaterals

28

5. All the diagonals from one vertex of a hexagon are drawn. How many triangles are formed inside?

Draw and verify your answer.

Answer

6. All the diagonals from one vertex of an octagon are drawn. How many triangles are formed inside?

Draw and verify your answer.

Answer

29

Polygons and Quadrilaterals

7. The sum of the interior angles of a convex polygon is 2340°.

How many sides does it have?

Answer

8. A convex polygon has 21 sides. Find the sum of its interior angles.

Answer

Polygons and Quadrilaterals

30

9. In a parallelogram, one side is 9 cm longer than the other.

If its perimeter is 66 cm, how long are its sides?

Answer

10. Which of the following shapes is a trapezium? Tick it.

11. Which of the following shapes is NOT a trapezium?

Tick it.

31

Polygons and Quadrilaterals

12. TMNK is a trapezium. Find the measures of angles M and K. M

136°

44°

T

N

Answer

K

13. In which pictures are the non-parallel sides of the trapezium highlighted in blue? Tick them.

B

A

C

D

14. The diagonals of a rhombus are 10 cm and 24 cm long.

Find the area of the rhombus.

C

B

Answer A

D

Polygons and Quadrilaterals

32

4. BAR DATA GRAPHS AND HISTOGRAMS HANDLING FREQUENCY DISTRIBUTION OF GROUPED AND UNGROUPED DATA 1. What is the lower limit of each interval? Circle the answer.

Intervals:

A.

1-4

1

0

2

B.

10-20

9

10

11

C.

21-35

20

22

21

D.

50-100

51

50

49

2. What is the upper limit of each interval? Circle the answer.

Intervals:

A.

2-5

6

5

4

B.

10-19

21

20

19

C.

25-36

36

35

37

D.

100-199

198

100

199

3. What is the size of each interval? Write your answers in the blanks.

33

A.

0-5

B.

1-10

C.

7-12

D.

20-24

E.

50-75

F.

240-260

Bar Graphs and Histograms

4. What is the class mark of each interval? Write your answers in the blanks.

A.

1-7

B.

30-45

C.

51-70

D.

60-98

5. The table shows students' marks in a geography exam.

Use this data to complete the frequency distribution table.

56

89

62

76

91

67

74

81

59

55

75

83

98

90

82

59

98

78

86

59

66

81

90

95

67

56

76

83

74

72

93

75

55-59

60-64

Class intervals Tally marks Frequency

6

6. The table shows the marks scored by 10 students in different subjects. Find the range of the marks in each subject.

Subject

Marks scored by 10 students

Range

Mathematics

67,87,58, 93, 54, 97, 56, 74, 61, 55

97 ­– 54 = 43

Science

84,52,49,88, 63, 73, 89, 47,90, 86

English

68, 54, 92, 80, 46, 65, 55, 48, 81, 70

Bar Graphs and Histograms

34

7. The number of floors in 20 buildings in a city are given below. Use it to complete the frequency table and find the range.

24, 27, 10, 16, 26, 27, 21, 25, 10, 13, 27, 29, 18, 10, 26, 10, 27, 18, 16, 21 Number of floor

10

Frequency

4

The range

8. The picture shows the number of flats in different towers of a society. Complete the frequency table and find the range of the data.

10

14

16

15

17

14 10

10 2

The range

35

Bar Graphs and Histograms

16

14

9. A store records the number of various types of chocolates sold over seven days. The data is shown below. Use it to answer the questions that follow.

Milk chocolate: 45, 62, 48, 50, 47, 55, 59 Dark chocolate: 32, 48, 35, 36, 39, 44, 37 White chocolate: 38, 40, 39, 47, 51, 45, 42 Nut chocolate: 65, 70, 58, 67, 59, 62, 61 А. Fill in the table.

Chocolate

30-34

Milk

35-39

40-44

45-49

50-54

55-59

60-64

3

1

2

1

65-70

Dark White Nut

B. Determine the range of sales for each type of chocolate.

1. Milk 2. Dark 3. White 4. Nut

C. Tick the chocolate that sold the most and circle one that sold the least.

1. Milk

3. White

2. Dark

4. Nut

D. Calculate the average daily sales for each type of chocolate.

Which type has the highest average sales?

1. Milk 2. Dark 3. White 4. Nut

Bar Graphs and Histograms

36

10. Eva took part in a spelling challenge and earned 113 stars.

Based on the table below, what is her rank?

Rank

Number of stars

Iron

0-12

Bronze

13-26

Silver

27-50

Gold

50-76

Platinum

77-100

Diamond

101-130

Ascendant

130-159

Immortal

160-200

Radiant

200 and more

The range

11. A store is offering discounts based on the number of products purchased. Use it to answer the given questions.

1-7 products

1%

46–60 products

7%

8–15 products

2%

61–75 products

10%

16–30 products

3%

76–90 products

15%

31–45 products

5%

91+ products

How much discount will a buyer receive if he purchases 31 products?

What is the minimum number of products required to receive a 15% discount?

37

Bar Graphs and Histograms

20%

12. A group of 11 people wants to go to an amusement park. Together, they weigh 767 kg. The shortest person in the group is 143 cm tall. How many rides can they all go on together?

Name of ride

Number of people

Maximum weight (kg)

Minimum height (cm)

Merry-go-round

1-5

300

120

Carousel

1-10

1000

150

Roundabout

4-12

1000

140

Flip-flap

1-10

1000

130

Tilt-a-Whirl

2-15

1000

140

Top spin

4-15

2000

140

Wave swinger

4-8

800

150

Rotor

4-10

2000

140

Answer

13. A girl wants to make pizza. She has 200 g of flour and 340 g of cheese. The table shows the amount of these ingredients needed for each type of pizza. How many types of pizza can she make?

Pizza name

Flour (g)

Cheese (g)

Margherita

180-200

300-320

Four Cheese

160-180

500-520

Chicken

140-160

100-120

Hawaiian

180-200

120-140

Seafood

220-240

100-120

Pepperoni

210-230

250-270

Mushroom

220-240

150-170

Veggie

150-170

200-220

Answer

Bar Graphs and Histograms

38

GRAPHICAL REPRESENTATION OF UNGROUPED DATA 14. The bar graph shows the average lifespan of five animals. Read the graph and answer the question.

Lifespan (years) 25 20 15 10 5 Rabbit

Hen

Wolf

Dog

Animals

Cat

How many years does a wolf live? Tick the correct answer.

16

17

18

19

15. The bar graph shows the lengths of different sports courts and fields. Read the graph and answer the question below.

Length (in metres) 100 80 60 40 20 Sports

What is the length of the basketball court? Tick the correct answer.

32

39

Bar Graphs and Histograms

22

28

50

16. The chart shows how many hours a day different animals sleep. Read the graph and answer the question below.

Hours of sleep per day 30 20 10

Mouse

Seal

Koala

Animals

Monkey

Which animal sleeps the least? How many hours does it sleep a day?

9

4

6

10

17. The bar graph shows the number of tourists who visited Chennai in various months. Read the graph and answer the question below.

400

Number of tourists

300 200 100 Mar.

Apr.

May

Jun.

Jul.

Months

Aug.

In which month did the fewest tourists visit? How many tourists visited that month? Tick the correct option.

210

235

260

201

Bar Graphs and Histograms

40

18. Read the data given in the table. Use it to draw the missing bar on the graph.

Lifespan (in years) 30

Animals

Lifespan (in years)

Rabbit

5

Cat

10

Horse

20

Lion

15

20 10

Rabbit

Cat

Horse

Lion

Animal

19. The table shows the amount of crop harvested (in tonnes) from three fields over four days. The data for Field 3 is shown in the bar graph. Draw the missing bar.

40

Crop harvested (in tonnes)

32 24 16 8 Day 1

41

Day 2

Day 3

Day 4

Days

Day 1

Day 2

Day 3

Day 4

Field 1

8 tonnes

12 tonnes

5 tonnes

11 tonnes

Field 2

12 tonnes

14 tonnes

7 tonnes

8 tonnes

Field 3

16 tonnes

12 tonnes

28 tonnes

32 tonnes

Bar Graphs and Histograms

20. The chart shows the amount of rainfall (in mm) in Mumbai for each month of the year. Use it to answer the given question.

Rainfall (mm) 75 50 25 Jan.

Feb. Mar. Apr. May Jun.

Jul.

Aug. Sep. Oct. Nov. Dec.

Months

How much rainfall was recorded in May? Tick the correct option.

58 mm

48 mm

54 mm

50 mm

21. The chart shows the heights of four famous towers.

Use it to answer the given question.

Height (m) 100 75 50 25 Pisa

Spassky

Big Ben

Belem

Tower

What is the height of the Spassky Tower?

96 m

78 m

71 m

60 m

Bar Graphs and Histograms

42

22. The table shows how many gadgets were sold from Monday to Thursday. Use the data from the table to complete the missing bars on the chart.

Number of phones sold 140 120 100 80 60 40 20 Mon

Tue

Wed

Thu

Gadget type

Mon

Tue

Wed

Thu

Tablets

24

26

38

18

Phones

20

70

30

50

Laptops

22

32

26

19

Days

23. Draw the missing bar to complete the graph.

The table shows the number of plants with flowers and without flowers in three seasons.

The bar graph shows how many more plants had flowers than no flowers in each season.

Number of plants 15 12 9 6 3

Spring

43

Summer

Bar Graphs and Histograms

Autumn

Spring

Summer

Autumn

Plants with flowers

46

59

36

Plants with no flowers

34

44

30

Seasons

24. Use the table to draw the missing bar on the graph.

The table shows the amount of apples harvested by four farmers over three days.

The bar graph shows the amount of apples harvested together by the 2nd and 4th farmers.

Apples harvested (kg)

Day 1 (kg)

Day 2 (kg)

Day 3 (kg)

Farmer 1

45

55

50

Farmer 2

25

40

25

Farmer 3

15

35

40

Farmer 4

20

15

25

60 50 40 30 20 10 Day 1

Day 2

Day 3

Days

25. Use the table to draw the missing bar on the graph.

The table shows the number of books of four genres sold by an online store over three days.

Number of books sold

Fiction

400 300 200 100

Fiction Mystery Comics Science

Mystery Comics Science

Monday

30

100

50

50

Tuesday

50

100

100

50

Wednesday

70

50

150

100

Book genres

Bar Graphs and Histograms

44

26. The bar graph shows the heights of four towers. Read the graph and decide if the given statements are true or false. Tick the correct option.

Names of the towers Tower of Belem Big Ben Spassky Tower Tower of Pisa 20

40

60

80

100

Height (in metres)

A. The Tower of Pisa is taller than the Spassky Tower.

True

False

B. The Spassky Tower is the shortest tower shown.

True

False

27. The bar graph shows the wingspans of four birds.

Read the graph and decide if the given statements are true or false. Tick the correct option.

Bird names Vulture Marabou Albatross Condor 50

45

100

150

200

250

300

350

400

Wingspan (cm)

A. The albatross has a wingspan 50 cm larger than the vulture.

True

False

B. The marabou has the smallest wingspan.

True

False

C. The albatross has a wingspan 50 cm larger than the vulture.

True

False

Bar Graphs and Histograms

28. The bar graph shows the lengths of three fish species.

Use the information below to fill the missing labels for the columns.

Length (m)

12 8 4

Fish

Maximum length

Whale shark

13 m

Beltfish

7m

Blue marlin

5m

Fish species

Beltfish

Whale Shark

Marlin

29. The table shows the names and heights of the tallest towers in three cities. Use it to label each bar in the graph with the correct city name.

Height of the tallest tower (m) 900 750 600 450 300

Tower

Height

Burj Khalifa in Dubai

828 m

Radio Tower in Tokyo

634 m

Ostankino Tower in Moscow

150

540 m

Cities

Bar Graphs and Histograms

46

Graphical Representation of Grouped Data 30. The bar graph shows the average long-jump results of two athletes in June, July and August. Read the graph carefully and answer the questions.

Jump length (cm)

Athlete 1

250

Athlete 2

200 150 100 50

June

July

Months

August

A. What was the long-jump result of Athlete 2 in June?

B. What was the long-jump result of Athlete 1 in July?

C. What were the results of Athlete 1 and Athlete 2 in August?

31. The chart shows how many adults and children visited the pool each month. Use the data to complete the table.

Month

Number of visitors

Child

January

2000 1800 1600 1400 1200 1000 800 600 400 200

February March April Adult

January

47

Adult

February

Bar Graphs and Histograms

March

April

Months

Child

32. The chart shows the number of books issued by a library to adults and children across five days. Fill in the table using the graph. Then answer the questions.

200 180 160 140 120 100 80 60 40 20

Number of books issued

Mon

Day

Adult

Wed

Tue

Thu

Adult

Child

Fri

Child

Days of the week Total

Monday Tuesday Wednesday Thursday Friday A. On which days was the maximum number of books issued in all?

B. What is the total number of books issued in the week?

C. Which day showed the highest demand for books?

D. How many more books were issued on Friday than on Monday?

E. On which day was the difference between the number of books issued to adults and children the smallest?

Bar Graphs and Histograms

48

33. The chart shows the quarterly sales of a company in 2023 and 2024. Use the graph to complete the table.

Then answer the questions that follow.

Sales (million ₹) 60

2023

50

2024

40 30 20 10 1 quarter

Year

2 quarter Sale in Q1 (million ₹)

3 quarter Sale in Q2 (million ₹)

4 quarter Sale in Q3 (million ₹)

Quarters Sale in Q4 (million ₹)

2023 2024 A. What is the average sales per quarter in 2023?

B. What is the average sales per quarter in 2024?

C. What is the total sales amount in 2023?

D. What is the total sales amount in 2024?

E. What is the average sales per quarter across both years combined?

F. What is the ratio of quarters with above-average sales to those with below-average sales?

49

Bar Graphs and Histograms

mln, ₹

mln, ₹ mln, ₹ mln, ₹ mln, ₹ mln, ₹

34. The table shows the number of students grouped by how many books they read over the summer. Draw a histogram for the given data.

Number of students

Books read

Class intervals (books)

5-10

10-15

15-20

20-25

25-30

Number of students

7

10

9

5

3

35. The table shows how long students took to run 50 metres. Draw a histogram for the given data and label both axes.

Class intervals

Number of students

15-20

20-25

25-30

30-35

35-40

3

4

6

7

4

Bar Graphs and Histograms

50

36. The two charts show the number of buildings taller than 100 m in City A and City B. Study both charts. Then read each statement and decide if it is True or False. Tick the correct option.

Height of buildings (m)

100–110

110–120

120–130

130–140

140–150

150–160

Number of buildings above 100 m in the city A. Number of buildings 80 70 60 50 40 30 20 10 Height range (m)

Number of buildings above 100 m in the city B.

80

Number of buildings

70 60 50 40 30 20 10 Height range (m)

51

Bar Graphs and Histograms

160–170

170–180

180–190 190–200

A. The city A has more buildings with a height of 100-110 m.

True

False

B. The city B has the most buildings with a height of 100-110 m.

True

False

C. There are more 140-150 m tall buildings in city A than in city B.

True

False

D. There are fewer 160-170 m tall buildings in city A than in city B.

True

False

E. There are 30 buildings in the city A with a height of 110-120 m.

True

False

F. There are 10 buildings in the city B with a height of 170-180 m.

True

False

G. There are more buildings in city A with a height of 120 to 130 m than in city B.

True

False

H. There are more buildings in city A with a height of 130 to 140 m than in city B.

True

False

I. There are no buildings between 190–200 m in city A.

True

False

J. The number of buildings 160-170 metres high in city A is equal to the number of buildings 130-140 m high in city B.

True

False

K. There are twice as many buildings in city A that are 150-160 m high as in city B.

True

False

Bar Graphs and Histograms

52

5. PIE CHARTS DATA HANDLING DRAWING AND READING PIE CHARTS 1. The pie chart shows the number of species belonging to different orders of mammals. Which order of mammals has the fewest species? Tick the correct option.

Insect eating Primates

Predators Rodents

Insect eating

Predators

Primates

Rodents

Bats

2. The pie chart shows the amount of rainfall in various months in Melbourne, Australia. Which month has the highest rainfall? Tick the correct option.

April May June August

53

July

March

May

June

April

August

Pie Charts

3. The pie chart shows the area occupied by the world’s largest islands. Which island is larger than Madagascar but smaller than New Guinea? Tick the correct option.

Greenland Borneo New Guinea Madagascar

Greenland

Borneo

New Guinea

Madagascar

4. The pie chart shows the recommended proportions of components of food in a diet. Is it true that fruits and vegetables should be eaten four times more than proteins? Tick the correct option.

True False Not enough information

Fruits and vegetables

Proteins

Carbohydrates

Fats

Pie Charts

54

5. The pie chart shows how some large volcanoes on Earth are distributed across the continents. How many volcanoes are there in North and South America altogether? Write the answer in the blank.

Eurasia

North America 7 volcanoes

20 volcanoes

Africa 3 volcanoes South America 8 volcanoes

Answer:

volcanoes.

6. The pie chart shows the amount of different nutrients in a 100-gram piece of cake. How much fat is there in the piece of cake? Write the answer in the blank.

Water 14 g

Carbohydrates 43 g

Fats ?

Proteins 18 g Answer:

55

Pie Charts

grams.

7. The pie chart shows the number of satellites of the giant planets in the Solar System. How many more satellites does Jupiter have than Uranus? Write the answer in the blank.

Saturn

Uranus

27 satellites

62 satellites

Neptune

14 satellites

Jupiter

67 satellites

satellites.

Answer:

8. The pie chart shows how the seas are distributed among the four oceans. How many seas do not belong to the Atlantic Ocean? Write the answer in the blank.

The Pacific

The Indian 6 seas

31 seas

The Atlantic 28 seas

The Arctic 15 seas

Answer:

seas.

Pie Charts

56

9. The table shows how the countries of the world are distributed across the continents.

Study the table. Then tick the pie chart that correctly represents this data.

Continents

% of countries

Eurasia

50.5%

North America

12.6%

South America

6.7%

Africa

30.2% Eurasia North America South America Africa

10. The table shows the annual rainfall in different natural zones. Study the table. Then tick the pie chart that correctly represents this data.

Natural zones

Rainfall (cm)

Forests

5.4

Prairie

3.0

Boreal forest

3.7

Cold desert

2.4

Prairie Forest Boreal forest Cold desert

57

Pie Charts

11. The table shows the area covered by the major oceans of the world. Study the table. Then tick the pie chart that correctly represents this data.

Ocean

Area (million km²)

Atlantic

92

Pacific

179

Indian

76

Arctic

15

Indian Atlantic Arctic

Pacific

12. The table shows the population of the four largest cities in Europe (2016). Study the table. Then tick the pie chart that correctly represents this data.

City

Rainfall (cm)

St. Petersburg

5,225,690

London

8,416,999

Moscow

12,330,126

Berlin

3,479,740

London Moscow St. Petersburg Berlin

Pie Charts

58

13. A survey was conducted in a class of 30 students to find their preferred destination for a school trip.

Study the data. Then draw and colour a pie chart to show the results.

Destination Option

Number of Students

Planetarium

12

Science Museum

9

Amusement Park

6

Nature Park

3

Remember to add a title (header) and a legend (key) to label each section.

Planetarium Science Museum Amusement Park Nature Park

59

Pie Charts

14. Steve wants to make a pie chart to show how he spends his 24-hour day.

He first recorded the data in the table shown.

Activity

Angle

Sleep

100°

School

70°

Homework

50°

Free time

70°

Eating and chores

60°

However, the angle measures he used are not correct.

Answer the questions to find and correct this mistake.

A. Add the angles. Do they make 360°?

B. Correct one or more angles so that the total becomes exactly 360°.

C. Draw a pie chart using your corrected angles. Label all sections clearly.

320°

340°

0°

20° 40°

300°

60° 80°

280°

100°

260°

120°

240° 220° 200°

180°

160°

Sleep School Homework Free time Eating and chores

140°

Pie Charts

60

6. PROBABILITY UNDERSTANDING PROBABILITY 1. Read each statement. Mark certain events with the letter C and impossible events with the letter I. There will be a full moon on 31st February.

The sum of the angles of a rhombus will be equal to 360 degrees.

Among lottery tickets numbered from 1 to 100, the winning ticket will be the one with number 127.

When rolling a fair die, a number from 1 to 6 will come up.

There exist two lines that intersect at more than one point.

2. An elementary event is an event with exactly one possible outcome. Read each statement. Which statements describe elementary events? Tick them.

A red ball will be drawn when taking a ball from a bag containing red and blue balls.

When flipping two fair coins, at least one of them will land heads up.

In a test with 15 questions, a student will answer no more than 7 questions correctly.

When rolling two fair dice, the first die shows 1 and the second die shows 6.

When randomly choosing two integers, the product of the selected numbers will be less than 0.

61

Probability

3. A fair die is rolled. What is the probability of rolling a 4?

Tick the correct answer.

4. Solve each of the following problems. Write your answers in the blanks.

A. A music teacher randomly plays one of seven musical notes. What is the probability that the note is C? B. A bag contains five balls of different colours: red, yellow, green, blue and orange. What is the probability of drawing the green ball?

C. A random experiment has exactly two possible outcomes. The probability of one outcome is 0.57.

Find the probability of the second outcome.

5. A football team has 11 players. Of these, 1 is a goalkeeper and 10 are field players. The coach randomly selects one player for an interview after the match.

What is the probability that the selected player is the goalkeeper? Tick the correct answer.

Probability

62

6. A box contains 35 marshmallows, of which 28 are chocolate-flavoured. A marshmallow is chosen at random. What is the probability that it is chocolateflavoured?

0.8

0.5

0.2

7. A mechanical clock stopped randomly. What is the probability that the hour hand stopped after 7 and before 1 on a 12-hour dial?

0.5

0.7

0.2

8. While baking a cake, a chef checks its weight.

The probability that its weight is less than 950 g is 0.99, and the probability that its weight is more than 900 g is 0.94. What is the probability that the cake’s weight is between 900 g and 950 g? Tick the correct answer.

0.55

0.93

0.06

9. A manufacturer tested its new printer by printing 200 copies of a page containing some text. Due to a printing fault, 56 copies came out blank. Use this scenario to complete the following sentences. .

The total number of trials in this experiment was

The number of trials where the event blank page occurred was The probability of getting a blank page was

63

Probability

.

.

10. A random experiment was conducted in which one marble was drawn and its colour was recorded. The marble was put back before the next trial. Study the table showing the results of 30 such trials and answer the questions that follow.

Colour

Blue

Black

Red

Green

Frequency

2

14

6

8

Find the experimental probability of drawing a red marble.

Answer: Predict the total number of the red marbles in the bags.

Answer:

11. The automated parking system at a mall records the colour of every car entering the parking area. On a day, the system shows that 80 cars have entered the parking area. Of these, 32 were white, and the rest were black.

Use this scenario to answer the following questions. a) What is the experimental probability that the next arriving car will be black?

0.6

0.32

b) How many white cars would you expect among the next 90 arrivals?

32

36

42

Probability

64

7. SQUARES AND SQUARE ROOTS SQUARE OF A NUMBER 1. Find the squares of the given numbers. Fill in the blanks.

x

–4

x²

–2

–3

–1

0

1

1

9

2

3

4

4

16

2. Simplify the expressions where needed. Check each result and then join it to the correct cart.

–(–16)

–4

0²

0

–3+5

(–3)²

Zero

Positive

65

Squares and Square Roots

Negative

3. Simplify the expressions where needed. Check the results.

Tick the negative numbers and circle the non-negative numbers.

83

0 6–7

– 4 × 25 0²

(–5)² 12⁴

4. Choose the correct words and complete the following sentences.

Positive

Negative

Non-negative

The square of a positive number is

Zero

. .

The square of a negative number is .

The square of zero is

.

The square of any rational number is

5. Which of the following numbers cannot be the square of any rational number? Tick the correct options.

1

0

– 100

256

25

– 64

1.44

9 100

Squares and Square Roots

66

6. Find the length of the side of the square shown. Then use the side length to find the area of the square. Write your answers in the boxes.

1m

Area =

m2

7. Find the value of each square.

67

Squares and Square Roots

m

8. Tick the correct statements.

A. The square of a positive number is negative.

B. 6 is greater than the square of 2.

C. The square of 3 is 9.

D. The square of a number is always less than the number.

9. Look at the first row. Observe the pattern used to get the second and third rows. Use the pattern to complete the table.

2

3

0

5

–1

10

4 9

1

25

10. Find the following products using the rule (a − b)(a + b) = a² − b². a) 82 × 78 b) 49 × 51 c) 63 × 57 d) 94 × 86 e) 25 × 35

Squares and Square Roots

68

SQUARE ROOTS 11. Look at the square shown below. What are the lengths of its sides? m

64 m²

m

12. Look at the square shown below. What are the lengths of its sides? m

121 m²

m

13. Look at the following expressions. Find the values of a, b and c.

69

Squares and Square Roots

14. Find the value of each of the following square roots.

15. Which number makes the following equation true?

Tick the correct option.

² = 2.25 1.5

2.5

15

– 0.5

16. Which number makes the equation true?

Tick the correct option.

² = 169 12

– 13

14

169

17. Which number makes the equation true?

Tick the correct option.

² = 0.0081 0.09

0.9

0.009

9

Squares and Square Roots

70

18. Find the square roots of the following decimal numbers. a) 5.0625

b) 33.64

c) 92.16

19. Point A on the number line represents one of the given numbers. Which number is it? Tick the correct option.

2

71

A

Squares and Square Roots

3

4

20. Point A on the number line represents one of the given numbers. Which number is it? Tick the correct option.

2

3

4

A

21. An electrician needs to lay a cable across the ceiling of a rectangular room from one corner to the opposite corner. The room measures 12 m by 5 m. What is the minimum length of cable required?

Squares and Square Roots

72

8. CUBES CUBE ROOTS DATAAND HANDLING CUBE OF A NUMBER 1. The width and height of a cube are 2 cm each.

What is its length?

Length =

2cm

cm

2cm

? cm

2. Find the volume of each box. Tick the box that is the most spacious. a)

3cm

5cm

3cm

4cm 5cm

3cm

5cm

4cm

4cm

b)

7cm 7cm

73

9cm 7cm

Cubes and Cube Roots

9cm

8cm 9cm

8cm

8cm

6cm 6cm

6cm

3. А wooden block was cut into smaller cubes with an edge length of 1 cm.

Write the dimensions of the small cube: A. Length

cm.

B. Width

cm.

C. Height

cm.

1cm

D. Count the number of unit cubes in each layer of the given block.

Complete the table.

Number of layers

Number of cubes

1 2 3

C

4 B

5 6

A

4. How many unit cubes make up the larger cube shown below? Write your answer in the box.

?

Cubes and Cube Roots

74

5. Fill in the blanks. a) 1³ + 2³ + 3³ + 4³ + 5³ = (1 + 8 +

+

+ 125) = 225 = (

)²

b) 1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (1 +

+

+

=

=(

)²

c) 1³ + 2³ + 3³ + 4³ + 5³ +

= (1 +

+ 125 +

+

= 289 =

=(

)²

+

6. Find the difference.

A. 35³ – 34³

B. 58³ – 57³

C. 92³ – 91³

75

Cubes and Cube Roots

+ + 27 +

+

7. Evaluate the given cubes.

A.

–9³ –17³ –5³ –100³

B.

C.

(1.6)³ (0.25)³ (1.42)³ (15.2)³

Cubes and Cube Roots

76

8. Find the prime factorisation of each number.

Use the factors to decide which numbers are perfect cubes. Tick the perfect cubes.

A. 12673 42875 8778

B. 1009875 238328 345

77

Cubes and Cube Roots

9. Find the smallest number by which the given number should be multiplied to get a perfect cube. In each case, find the cube root of the new number formed.

A. 108 B. 250

10. Find the smallest number by which the given number should be divided to get a perfect cube. In each case, find the cube root of the new number formed.

A. 1024 B. 2592

11. How many boxes of light bulbs measuring 3 cm × 4 cm × 2 cm can fit into a shipping container, which is a perfect cube with sides of 50 cm?

50cm

50cm

50cm

2cm 3cm

4cm

Cubes and Cube Roots

78

Cube Root of a Number 12. Look at the figures in each row. Which of these shows the cube of a number? Tick the correct figure.

A.

B.

C.

D.

79

Cubes and Cube Roots

13. Shade the unit cubes along one edge of each figure to show the cube root of given numbers. Write the cube root in the box. А. Cube root of 27 is

C. Cube root of 1000 is

.

.

B. Cube root of 125 is

.

D. Cube root of 343 is

.

Cubes and Cube Roots

80

14. Match the figures to complete a cube.

A.

1.

B.

2.

3.

C.

D.

81

Cubes and Cube Roots

4.

15. Evaluate the following cube roots.

A.

B.

C.

Cubes and Cube Roots

82

16. Evaluate the following cube roots.

A.

B.

17. Fill in the blanks.

83

Cubes and Cube Roots

18. Complete the prime factorisation of each number.

Then use it to find the cube root of each. Fill in the blanks.

A. 5832

C. 13824

B. 4096

2

5832

2

2

2916

2

4096

2

2

13824

2

3456

2

864

2

216

243 2

3 27

3

128

2

3 1 27 3

6 = 2³ × 3 = 2³ × 9³

=

=

(2 × 9)

(2 × 9)

= 18

3

2

=

2 1

1 =

=

=

=

= =

=

=

=

=

=

Cubes and Cube Roots

84

19. Find the cube roots of the following numbers using successive subtraction method.

A. 64

85

Cubes and Cube Roots

B. 512

C. 3375

20. Find the smallest number that must be subtracted from each number to make it a perfect cube. Also, find the cube root of the perfect cube so obtained.

A. 9520

B. 131

C. 5000

D. 1200

Cubes and Cube Roots

86

21. Find the smallest number that must be added to each number to make it a perfect cube. Also, find the cube root of the new number so formed.

A. 29650

B. 11925

C. 6500

C. 4448

87

Cubes and Cube Roots

22. A package has a volume of 39,304 cm³. Can it fit in a car trunk with a length of 1,040 cm, height of 480 cm, and width of 1,080 cm?

480 cm

1040 cm

1080 cm

23. A package has a volume of 10,648 cm³. Can it fit in a cubic container with a side length of 25 cm?

24. A cubic container has a volume of 64 m³. It is currently 75% filled with smaller cubic boxes, each with a side length of 1 m. How many more boxes are needed to fill the container completely?

Cubes and Cube Roots

88

9. PERCENTAGE AND ITS APPLICATIONS Percentage and Percentage Change 1. The cost price of a computer is ₹90,000. Calculate the profit for each selling price given below.

A. SP = ₹1,70,000

B. SP = ₹2,20,000

C. SP = ₹3,10,000

89

Percentage and Its Applications

2. The cost price of a grocery basket is ₹1,000. Calculate the loss for each selling price given below.

A. SP = ₹456

B. SP = ₹743

C. SP = ₹899

Percentage and Its Applications

90

3. Look at the cost price (CP) and profit percent for the cars shown below. Find the selling price of each car.

A. CP = ₹2,53,000; Profit 24%.

B. CP = ₹9,00,000; Profit 40%.

C. CP = ₹12,00,000; Profit 60%.

D. CP = ₹15,00,000; Profit 7%.

91

Percentage and Its Applications

4. A store sold a smartphone for ₹12,000. Find its cost price in each case.

A. The phone was sold at a loss of 7%.

B. The phone was sold at a loss of 26%.

C. The phone was sold at a loss of 57%.

5. Look at the cost price (CP) and selling price (SP) shown below. Match the pairs that give a profit of 5%.

CP A. 1020 B. 1100 C. 1000 D. 1200

SP 1. 1260 2. 1071 3. 1050 4. 1155

6. Look at the cost price (CP) and selling price (SP) shown below. Match the pairs that give a loss of 15%.

CP A. 3500 B. 3452 C. 3670 D. 3780

SP 1. 3213 2. 3119.5 3. 2975 4. 2934.2

Percentage and Its Applications

92

7. A shopkeeper sells two laptops. The first laptop is sold for ₹1,38,000 and the second for ₹51,000. On the first laptop, he makes a profit of 15%, and on the other, he incurs a loss of 15%. What is the cost price of each laptop, and what is his total profit or loss percentage?

8. A stationery store owner bought 200 pens at ₹2.5 per piece. Among these, 12 pens turned out to be defective and had to be discarded. The remaining pens were sold at ₹3.2 per piece. Calculate the percentage of profit or loss.

93

Percentage and Its Applications

9. When selling household appliances for ₹2,64,000, the seller incurs a loss of 12%. At what price should the equipment be sold to make a profit of 7%?

10. A baker made jam that cost her of ₹8,500. She sold half of the jam at a loss of 10%. She wants a total profit of 25%. At what profit percentage should she sell the remaining jam?

Percentage and Its Applications

94

11. The marked price of a TV is ₹2,40,000. Find the discount and the selling price for each discount rate offered below.

A. Discount 15%

B. Discount 8%

12. The discount offered on two printers is given below.

Find the selling price in each case if the marked price is ₹18,350.

A. Discount offered ₹1,290

95

Percentage and Its Applications

B. Discount offered ₹3,000

13. The discount offered on two telescopes is shown below. Find the percentage of discount in each case if the marked price was ₹32,000.

A. Discount offered ₹5370

B. Discount offered ₹3490

14. The price of a pair of headphones dropped by 20%.

On weekend sale, there was an additional 10% discount on the reduced price. If Sam bought the headphones for ₹5,760, what was their regular price?

15. A kettle was marked at one third of the actual price. Due to a festive sale, the seller gave another 20% discount. What is the total discount percentage and the regular price of the kettle if Robin bought it for ₹1,200, including all discounts?

Percentage and Its Applications

96

16. A dishwasher costs ₹56,000. A GST (Goods and Services Tax) of 23% is applicable on it. What is the total amount to be paid to buy the dishwasher?

76,730

42,750

68,880

17. Dan bought a laptop for ₹56,340 that included GST of 14%. Find the price of the laptop before the GST was added.

49,421

57,275

51,888

18. Akash bought a lawn mower for ₹1,32,750 that included GST of 7%. How much was the amount of GST, rounded to the nearest rupees?

97

9067

Percentage and Its Applications

7477

8685

19. A radio receiver cost ₹34,500 before tax and ₹37,260 after GST was applied. Find the percentage of GST applied.

7%

8%

9%

20. Ravi got a 15% discount on a computer with a marked price of ₹1,55,000. He had to pay tax on the discounted price. Calculate the final amount he would pay if the transaction were to occur in two different regions with the following tax rates: A. Tax of 12%

B. Tax of 9%

Percentage and Its Applications

98

21. A manufacturer sells wooden chairs at ₹4,280 per piece, including 7% GST. The manufacturer makes a profit of 25% on the production cost. Determine the production cost of one chair.

22. Mayank bought the following items for his new office: Office desk for 95,000 rupees Executive chair for 45,000 rupees Filing cabinets for 75,000 rupees All of these products are subject to a GST of 8%.

Calculate the total amount that Mayank had to pay for the entire purchase.

99

Percentage and Its Applications

23. The selling price of 20 cups of coffee without tax is equal to the selling price of 16 cups of coffee with tax.

What is the tax percentage?

24. A family wants to buy household appliances. Calculate the total purchase amount in both stores. Compare which store is more profitable if the tax charged on household appliances is 20%.

Price, ₹ Product

Store 1

Store 2

Refrigerator

45,000

37,000

Washing machine

64,000

69,000

Vacuum cleaner

37,500

29,500

Microwave oven

27,400

31,200

Store 1 Store 2

Percentage and Its Applications

100

10. COMPOUND INTEREST Simple Interest and Compound Interest 1. Fill in the table.

Principal

Rate p.a.

Time

A

₹550

6%

1 year

B

₹1200

C

₹1500

D

Simple interest

5 months

Amount

₹90

11%

₹330

7.5%

11 months

₹720

2. Matthew wants to earn at least ₹300 as simple interest in 3 years. Look at the principal amount and interest rate for each deposit option. Which deposit should he choose?

101

Deposit option

Principal

Rate p.a

Time

A

₹1850

5%

3 years

B

₹1600

7%

3 years

C

₹800

11%

3 years

Compound Interest

Simple interest

3. Amar plans to place a deposit of ₹15,000 in Summa Bank for a period of 2 years. The Bank offers an interest rate of 3.75% per annum under the simple interest rate.

Calculate: A. The amount of interest he will receive for the entire period.

B. The total amount in the account at the end of the deposit period.

4. Salim deposited ₹4,200 in the bank on February 15, 2023 at 8% per annum simple interest. Calculate the amount of accrued interest: A. By May 10, 2023 B. By July 17, 2023

Compound Interest

102

5. Find the amount and the compound interest on ₹98,500 for 4 years at 7% per annum, compounded annually.

6. Raj borrowed ₹9,000 and returned ₹10,300 after 3 years, along with accrued interest. What rate of simple interest did Raj pay?

103

Compound Interest

7. Which will earn more interest, and how much more?

A. ₹25,000 lent at 10% per annum compounded annually for 2 years.

B. ₹25,000 lent at 10% per annum compounded semi-annually for 2 years.

8. A bank offers four deposit options. In each option, an initial deposit of ₹2,00,000 is made for 3 years.

A. Calculate the final amount for each option.

B. Which option is most profitable?

A

6%

Annually

B

5.8%

Semi-annually

C

5.7%

Quarterly

D

5.6%

Monthly

Compound Interest

104

9. A small resort town has a population of 850,000 that is growing at an annual rate of 8%. What will the population be in 2 years?

A. 986440

C. 991440

B. 891440

D. 901440

10. Amit bought commercial real estate for ₹30,00,000.

Market analysts predict that prices for such properties will increase at 7% annually. What will the property be worth in 4 years?

105

A. 38,32,388

C. 39,32,388

B. 37,32,388

D. 36,32,388

Compound Interest

11. A company bought musical equipment for ₹9,50,000 in 2021 and plans to sell it in 2024. The depreciation rate is 35% per year. At what price can the company sell the equipment in 2024?

A. 2,70,894

C. 2,60,894

B. 2,50,894

D. 2,80,894

12. Read the assertion and reason statements.

Tick the correct option.

Assertion (A): Priya invests ₹2,50,000 at 8% per annum compounded annually for 3 years. She will receive ₹3,14,928 after 3 years.

Yes No

Yes No

Yes No

Yes No

Reason (R): The amount after compound interest is calculated as principal multiplied by (1 + rate) raised to the power of time.

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

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

A is true, but R is false.

A is false, but R is true.

Compound Interest

106

11. ALGEBRAIC EXPRESSIONS

DATA HANDLING

Addition and Subtraction of Algebraic Expressions 1. Complete the table.

Expressions

Terms

Factors

Coefficient of x

Constant

A B C D

2. Look at the given algebraic tiles. Write the algebraic expression represented by each set of tiles. x

x

–x²

–x²

1 –x²

x

x

1

1

1

107

–x²

x

–x²

–x²

Algebraic Expressions

x

1

x

1

1

1

1

1

1

1

1

1

1

x

x

x

1

1

x

x

x

1

1

3. Write each expression in the correct column of the table.

Monomials

Binomials

Trinomials

Quadrinomials

Algebraic Expressions

108

4. Identify which expressions have like terms.

Join the pairs.

5. Match each verbal description to its algebraic expression.

A. The square of the difference between m and n, reduced by the tripled product of these variables.

1.

В. The sum of the square of x and the triple of y, increased by the product of x and y.

2.

C. The difference between triple the 'product of m and n' and quadruple m and n.

3.

D. The quotient of the 'sum of p and q' divided by triple the sum of these same numbers.

4.

E. The product of the 'sum of p and q' and their difference.

F. The quotient of the 'product of m and n' divided by triple the product of these same numbers.

109

Algebraic Expressions

5.

6.

6. Simplify. a)

b)

c)

d)

Algebraic Expressions

110

7. Align the expressions vertically by like terms.

Then simplify. a)

b)

c)

d)

e)

111

Algebraic Expressions

8. Simplify the expressions. a)

b)

c)

d)

9. Connect each equation to the expression missing in it. a)

1)

b)

2)

c)

3)

d)

4)

Algebraic Expressions

112

Multiplication of Algebraic Expressions 10. Multiply the polynomials.

a) (3 )( + 1) = b) (5 + 1)(7 ) = c) (− + 7)( + 4) = d) (3 )(6 + 3) =

11. Find the area of each rectangular painting. a)

113

Algebraic Expressions

b)

c)

d)

Algebraic Expressions

114

12. Find the area and verify the result for the given values of the variables. a)

b)

c)

d)

115

Algebraic Expressions

13. Simplify. a)

b)

c)

d)

Algebraic Expressions

116

14. A cuboid-shaped structure has dimensions that vary based on three variables. Its length is (x + y + z) metres, its width is (2x − y + 3z) metres, and its height is 5xy metres.

A. Write a formula for the volume of the cuboid. Simplify your answer.

B. Calculate the volume when x = 100, y = 50, and z = 200.

C. Calculate the volume when x = 200, y = 130, and z = 170.

117

Algebraic Expressions

15. A cuboid-shaped space station has length (3a + 2b) metres, width (a² − b + 5) metres, and height 4ab metres, where a represents the number of physics laboratories and b represents the number of biology laboratories. Each physics laboratory occupies 1000 m³ of space and each biology laboratory occupies 1500 m³ of space.

A. Write a formula for the total volume of the station. Simplify your answer.

B. Calculate the volume of the station when a = 5 and b = 3.

C. Using the values from part B, calculate the total volume occupied by all laboratories.

D. Find the volume remaining for living quarters and other facilities.

Algebraic Expressions

118

12. AREA OF POLYGONS Area of Figures Made with Polygons 1. A trapezium has bases 8 cm and 14 cm and height 5 cm. A semicircle is constructed on the longer base as its diameter. Find the total area. (Use π = 3.14.)

2. A right triangle with legs 6 cm and 8 cm is joined to a rectangle measuring 8 cm × 5 cm along the 8 cm leg. A semicircle is constructed on the triangle's hypotenuse. Find the total area. (Use π = 3.14.)

119

Area of Polygons

3. A rectangle with sides 10 cm and 6 cm has an isosceles triangle on top. The triangle has a base of 10 cm and height 4 cm. Find the total area.

4. A circle with diameter 12 cm is cut from a square with side 12 cm. Find the area of the remaining part. (Use π = 3.14.)

Area of Polygons

120

5. Find the approximate area of each shape.

The area of this square is 1 unit2.

6. Draw a rectangle around each triangle. Then estimate the area of the triangle by subtracting the extra regions from the rectangle's area.

The area of this square is 1 unit2.

121

Area of Polygons

7. Can the area of each shape be calculated with the given information? Tick Yes or No. Yes

Yes

No

Yes

No

25 m

No

30 m 20 m

35 m

10 m

25 m

35 m

15 m 15 m

5m

30 m

10 m

8. Now calculate the area of each shape in previous question for which you answered 'Yes'.

Area of Polygons

122

9. Find the area of each shape. Round each answer to the nearest whole number.

60cm 10cm

A

30cm

4cm

B

C

20cm 50cm 20cm

123

Area of Polygons

D

40cm 10cm

E

10cm

30cm

F

G 90cm

12cm

J

H 4cm

25cm

25cm

40cm

I

40cm

8cm

Area of Polygons

124

13. SURFACE AREA AND VOLUME OF SOLIDS SURFACE AREA 1. Find the surface area of a rectangular prism (cuboid) with length 6 cm, width 4 cm and height 3 cm. b

c

a

Answer

2. A cylinder has radius 2 cm and height 5 cm. Find: a) the lateral surface area; b) the total surface area (Use π = 22/7). r

h Answer

125

Surface Area and Volume of Solids

3. There are 30 identical circular discs. Each disc has radius 15 cm and thickness 1.5 cm. The discs are stacked to form a solid cylinder. Find the total surface area of the cylinder. Use π = 22/7.

Answer

4. The edge of a cube is 8 cm long. The diagram shows its net with 1 cm glue flaps on each joining edge. A cardboard sheet measuring 40 cm × 25 cm is available. Will the net fit on this sheet?

Surface Area and Volume of Solids

126

5. The volume of a cube is 125 cm³. Find the area of one of its faces.

125 m³

Answer

6. The volume of a cube is 1m³. Find its surface area.

1 m³ Answer

127

Surface Area and Volume of Solids

7. Four identical cubes, each with volume 8 cm³, are stacked to form a tower. Calculate the height of the tower.

8 cm³

Answer

8. A cube has volume 1000 cm³. A ladybug crawls along its edges, as shown in red. Calculate the total distance it crawled.

1000 cm³ Answer

Surface Area and Volume of Solids

128

14. EXPONENTS AND POWERS DATA HANDLING Exponents 1. Simplify the numbers on each pair of flasks.

Circle the greater value in each pair.

A.

6²

2⁶

9⁵

5⁹

4⁷

7⁴

B.

C.

129

Exponents and Powers

2. Express each number in expanded form using powers of 10.

A. 234.56 B. 7089.004 C. 0.12345 D. 6000.0078 E. 9876.54321

For each number, determine the sum of all the coefficients.

A. 234.56 B. 7089.004 C. 0.12345 D. 6000.0078 E. 9876.54321

Which number has the most significant figures?

Which numbers have the highest power of 10?

Exponents and Powers

130

3. Express each fraction using negative exponents. b)

c)

d)

e)

4. Express each of the following as a fraction.

a)

b)

c)

d)

e)

5. Express each of the following as a fraction. a)

131

b)

Exponents and Powers

c)

6. Write each of the following without exponents. a)

b)

c)

d)

7. Simplify.

a)

d)

c)

b)

8. By what number should 1 4

4

be divided to get 25?

16

9. By what number should 1 9

81

1 16

be multiplied to get 225?

9

1 81

Exponents and Powers

132

10. Write the following large numbers in standard form.

Then compare them.

A.

56000000

570000000

9500000

890000000

320000000

700000000

B.

C.

133

Exponents and Powers

11. Simplify each expression by applying the rule

a n an = n. b b

Exponents and Powers

134

12. Simplify the expressions.

13. Simplify each expression. Write the missing numbers.

A.

B.

C.

135

Exponents and Powers

14. Solve the following equations.

15. Simplify the following expressions.

Exponents and Powers

136

15. DIRECT AND INVERSE PROPORTIONS PROPORTIONS 1. Find the ratio of the number of floors in the blue building to the number of floors in the pink building.

:

2. Use your answer from the previous question to fill in the blanks.

:3= The blue building has

3

=

times as many floors as the pink building.

3. Write the ratio of the number of motorcycles to the number of cars.

:

137

Direct and Inverse Proportions

4. Find the ratio of the number of floors in the blue building to the number of floors in the pink building.

:

5. Use your answer from the previous question to correct the error in the following equation. Write your answer as a decimal number.

7:4=

3 = 4

6. Circle the picture that matches the given statement.

1 The number of flowers is 4 of the number of cacti.

Direct and Inverse Proportions

138

7. For each case, choose the picture that matches the description. Write the correct letter in the blank.

B

A

C

Case1. There are 3 times as many glasses with orange juice as with apple juice.

Case 2. There are 2 times as many glasses with orange juice as with apple juice.

8. Choose the correct ratio for each case from the previous question.

3:1 Case 1.

3:2

:

2:8

1:2 Case 2.

2:1

:

9. Which ratios correctly describe the relationship between glasses of orange juice and glasses of apple juice shown in the image? Tick the correct answers.

10. Which ratios correctly describe the relationship between glasses of orange juice and glasses of apple juice shown in the image? Tick the correct answers.

139

Direct and Inverse Proportions

11. Find the ratio of the weight of the burger to the weight of the fries.

60 g 166 Cal

300 g 800 Cal

:

12. Use the ratio from the previous question to fill in the blanks.

: 3 = The burger is

=

times heavier than the fries.

13. Find the ratio of the weight of the burger to the weight of the ice-cream.

300 g 800 Cal

100 g 164 Cal

:

14. Use the ratio from the previous question to fill in the blanks.

: 3 = The burger is

3

=

times heavier than the ice-cream.

Direct and Inverse Proportions

140

15. Look at the picture. For each shelf, choose the correct ratio of bagels to cupcakes.

2:4

1:4

1:3

2:3

16. What is the ratio of the Sphinx's height to the pyramid's height? Tick the correct option.

139

139 : 20

20 : 139

20

Pyramid

Sphinx

17. The ratio of the panda's weight to the koala's weight is 137 : 15. Which statement can be determined from this ratio? Tick the correct answer.

141

The panda is

times heavier than koala.

The panda is

kg heavier than koala.

Direct and Inverse Proportions

18. Check for direct proportionality and complete the table.

Look at Table 1. a) Are the quantities x and y directly proportional? b) If yes, what is the constant of proportionality?

Yes

No

k=

x

5

15

25

35

y

10

30

50

70

c) The variables x and y in Table 2 are directly proportional, and the constant of proportionality is 2.5. Write the missing values in the table.

x

4

?

16

?

y

?

15

?

35

19. Check for inverse proportionality and complete the table.

Look at Table 1. a) Are m and n inversely proportional?

Yes

b) If yes, what is the constant of proportionality?

No k=

m

4

6

12

24

n

30

20

10

5

c) The variables p and q in Table 2 are inversely proportional. When p = 8, q = 9. Use this information to write the missing values in the table.

p

8

?

6

?

q

9

12

?

18

Direct and Inverse Proportions

142

20. Look at the numbers in each table and answer the given questions.

A. Are the quantities a and b directly proportional? Yes If yes, what is the constant of variation?

k=

a

7

21

35

49

b

36

108

190

282

B. Are the quantities a and b directly proportional? Yes If yes, what is the constant of variation?

No

No

k=

a

14

56

84

112

b

12

48

72

96

21. Complete the table if x and y vary in direct proportion.

x

33

y

12

132

176

55

4

22. Manish can type 960 words in 15 minutes. How many words will he type in 2 hours?

Answer

143

Direct and Inverse Proportions

23. A cyclist covers 72 kilometres in 4 hours. How many kilometres will he cover in 7 hours at the same speed?

Answer

24. Ravi can complete a job in 20 minutes. He works alone for 5 minutes. Meena then takes over and finishes the remaining work in 15 minutes. How long would it take if Ravi and Meena worked together from the start?

Answer

25. Pipe X can fill a pool in 24 hours. Pipe Y can fill the same pool in 12 hours. Pipe Z can empty the full pool in 48 hours. If all three pipes are opened simultaneously, how long will it take to fill the pool?

Answer

Direct and Inverse Proportions

144

16. FACTORISATION AND DIVISION OF ALGEBRAIC EXPRESSIONS Factorisation of Algebraic Expressions 1. What is the greatest factor common to all the terms in each expression? Tick the correct option.

2. Find the greatest factor common to all the terms in each expression. Write it in the blank.

3. Find the common binomial in each expression. Use it to factorise the expression.

145

Factorisation and Division of Algebraic Expressions

4. Factorise.

5. A rectangular plot has width x metres and length (x + 4) metres. A fence is built around the perimeter, and two additional partitions parallel to the width are installed inside to divide the plot into three equal parts. a) Write an expression for the total length of fencing needed. b) Simplify the expression and write it as a polynomial.

c) What is the greatest factor common to all the terms? Use it to factorise the expression.

Factorisation and Division of Algebraic Expressions

146

6. Factor each expression by regrouping the terms.

1) 2) 3)

7. Express each trinomial as the square of a binomial.

147

1)

3)

2)

4)

Factorisation and Division of Algebraic Expressions

8. Complete each trinomial so that it becomes a perfect square. Write the missing term.

1) 2) 3)

9. Find the value of the following expression for a = 4.

10. Find the value of the following expression for a = 3.

11. Find the value of the following expression for a = 1.

Factorisation and Division of Algebraic Expressions

148

12. Fill in the blanks.

1) 2) 3)

13. Evaluate each expression using the difference of squares formula.

1) 2) 3)

149

Factorisation and Division of Algebraic Expressions

14. What is the correct factorisation of the following polynomial? Tick the correct option.

15. Divide each monomial by the given monomial. Simplify your answers.

Factorisation and Division of Algebraic Expressions

150

16. Divide each polynomial by the given monomial.

1) 2) 3)

151

Factorisation and Division of Algebraic Expressions

4) 5)

Factorisation and Division of Algebraic Expressions

152

17. LINEAR GRAPHS DATA HANDLING Line Graphs and Linear Graphs 1. The graph shows the water level in river Ganga during major floods. During the 1986 flood, the water rose to 2.6 m.

Mark this point on the graph. Then join all the points to complete the line graph.

Water level (m) 2.8 2.6 2.4 2.2 2 1726 1752 1898 1977 1986

Year

2. The table shows Anusha's height as she grows. Plot all the given data points on the graph and draw the line to complete it.

Height (m) 1.3 1.25 1.2 1.15 1.1 1.05 1 4

153

Linear Graphs

5

6

7

8

Age (years)

Age (years)

Height (m)

4

1

5

1.1

6

1.15

7

1.2

8

1.25

3. The table shows sleep hours needed for the children of different ages. Plot all the given data points on the graph and draw the line to complete it.

Sleep (hours) 16 14 12 10 8 6 1

5

9

13

17

Age (years)

Sleep (hours)

1

14

5

12

9

10

13

9

17

8

Age (years)

4. The table shows the amount of rainfall in a city over five months. Plot all the given data points on the graph and draw the line to complete it.

Rainfall (mm) 24 18 12 6

Jun.

Jul.

Aug.

Sep.

Oct.

Month

Month

Rainfall (mm)

June

6

July

12

August

15

September

18

October

9

Linear Graphs

154

5. The graph shows how the price of sandals changed over several months. Use the graph to find how much the sandals cost in September.

Price (in ₹) 700

Answer:

600 500 400 June

July

Aug. Sept. Oct.

Month

6. The graph shows how a pillar's shadow length changed throughout the day. Use the graph to find the length of the shadow at 12:00. Tick the correct option.

Shadow length (cm) 175 155 135 115 95 75 55 9:00

85 cm

155

Linear Graphs

10:00

11:00

55 cm

12:00

75 cm

13:00

40 cm

Time (hours)

7. The graph shows how the helicopter’s altitude changed during the flight. How many minutes after the flight began did the helicopter reach the highest altitude?

Tick the correct option.

Height (m) 1000 800 600 400 200 10

20 min

20

30

40

30 min

50

40 min

Time (minutes)

60

50 min

8. The graph shows the water level in river Ganga at a place over several months. What is the highest water level shown on the graph? Tick the correct answer.

Water level (m) 10 8 6 4 2 April May June

8m

9m

July

Aug. Sept. Oct.

7m

Month

10 m

Linear Graphs

156

9. The graph shows the population of France in the first half of the 20th century. Find whether each statement is true or false. Tick the correct option.

Population (million people) 46 44 42 40 38 1910

1920

1930

1940

Year

1950

A. In 1930, the population of France was 45 million people.

True

False

B. The population of France was lowest in 1920.

True

False

C. The population of France in 1910 and 1930 was the same.

True

False

10. The graph shows water temperature in the Black Sea over 5 months. The Mediterranean Sea is 5°C warmer than the Black Sea in July. What is the water temperature in the Mediterranean Sea in July?

Water temperature (°C) 26 24 22 20 18 May

Answer:

157

Linear Graphs

June

July

Aug.

Sept.

Year

11. The graph shows bus waiting times at a stop for some hours of the day. Tram waiting time is 5 minutes shorter than bus waiting time. How long do you wait for a tram at 17:00?

Waiting time (minutes)

40 30 20 10 13:00

14:00

15:00

16:00

18:00 Time (hours of the day)

17:00

Answer:

12. The graph shows how a helicopter's altitude changed during its flight. Find all the hours when the helicopter was above 900 m.

Altitude (m) 1500 1300 1100 900 700 1

2

3

4

5

6

7

Time (hours)

Answer:

Linear Graphs

158

13. The graph shows how Earth's population changed in the second half of the 20th century. By how much did Earth's population increase from 1960 to 2000? Tick the correct option.

Population (million people) 7000 6000 5000 4000 3000 2000 1950 1960 1970 1980 1990 2000

2000 million

3000 million

4000 million

Year

5000 million

14. The graph shows the distance travelled by a car over time during its journey between two cities. How many kilometres did the car travel from 14:00 to 16:00? Tick the correct option.

Distance (km) 300 250 200 150 100 50 12:00 13:00 14:00 15:00 16:00 17:00

50 km

159

Linear Graphs

75 km

100 km

125 km

Time (hours of the day)

15. The graph shows a bus journey from Delhi to Ambala and back. Read each statement. Is it true or false? Tick the correct option.

Distance from Delhi (km) 200 175 150 125 100 75 50 25 0

1

2

3

4

5

6

7

Time (hours)

A. After 2 hours of travel, the distance from Delhi was 25 km more than after 5 hours.

True

False

B. From the first to the fifth hour of the journey, the distance from Delhi True only increased.

False

16. The graph shows the water level in a river near a city over several months. Decide whether each statement is true or false. Tick the correct option.

Water level (m) 1 0.8 0.6 0.4 0.2 April May June July Aug. Sept. Oct.

Month

A. The water level in the river in April and October was the same.

B. The lowest water level in the river was in May.

True

True

False

False

Linear Graphs

160

17. The graph shows how Earth's population changed in the second half of the 20th century. Decide whether each statement is true or false. Tick the correct option.

Population (billion people) 6 5 4 3 1950 1960 1970 1980 1990 2000

Year

A. Earth's population grew by 0.8 billion from 1970 to 1980.

True

False

B. From 1980 to 2000, Earth's population continued to grow.

True

False

18. The graph shows the number of people who visited a website each month. Decide whether each statement is true or false. Tick the correct option.

Visitors (million people) 24 22 20 18 16 Sep. Oct. Nov. Dec. Jan.

161

Months

A. The number of visitors increased from November to January.

True

False

B. In October, the site was visited by more people than in any other month.

True

False

Linear Graphs

19. The graph shows air temperature at midnight (0:00) and noon (12:00) for three consecutive days. Decide whether each statement is true or false. Tick the correct option.

Temperature (°C) 16 14 12 10 8 6 4 0:00

12:00

0:00

12:00

0:00

12:00 Time (hours)

A. It was 3 °C colder on first day at 12:00 than on the second day at the same time.

True

False

B. The lowest temperature was on the second day at 0:00.

True

False

20. The graph shows the height of a eucalyptus tree at different ages. How tall is the tree when it is 25 years old?

Height (m) 90 60 30

0

5

10

15

20

25

30

Age (years)

Answer:

Linear Graphs

162

ANSWERS

1. Rational Numbers (p. 1–16) 6.

1.

7.

0

2.

1

8.

3.

4.

Equal denominators:

9.

0

Different denominators:

1

10.

1. is divisible by 3 as well as 5.

5.

2. is divisible by 9 as well as 4.

3. is divisible by b as well as d.

11.

163

12.

18.

(*) possible answer

(*) possible answer

19.

32 8 5

10 2

20.

= 4; =

14

=

3

28 6 1 = ; 6 3 2 1 = ; 10 5 20 4 = ; 25 5 2 3 1 = = ; 8 12 4 9 1 = ; 99 11 7 = 1; 7 32 8

,

1 4

,

=

2 4

=

1 2

;

21. Blue car travels the farthest.

1 pizza was left.

3

22. 6, 4, 23.

9 8

a b

13. Correct solutions are:

× ×

d c

24.

—

14.

17.

15.

16.

—

25.

26.

164

4.

27.

True True True

False 28.

5.

3 min

30 min

15 min

The pool will be completely filled.

29.

x=3

x = –2 6.

30.

7.

a) Subtract a from both sides; x = b – a b) Step 1: Subtract 2 from both sides; Step 2: Divide both sides by (–5); x = –2

2. Solving Equations in One Variable (p. 17–26)

8.

1.

x = 9; x = Star (*) possible answer

2.

1 variable: more than 1 variables: 3. ax = b ax + b = 0 other

165

c) Step 1: Subtract 4 from both sides; Step 2: 3 16 Divide both sides by ;x= 4 3 5 7 ;x= or 3.5 6 2

9. one root

infinite roots

no roots

no roots

10.

x=3 x=1 11.

12.

3. a = 0? yes

no

trapezium b = 0? no

no roots

yes

x — any number

13.

4. trapezium

at least one pair of parallel sides

quadrilateral

any 4 sided polygon

rectangle

2 pairs of parallel sides equal angles, 2 pairs of parallel and equal sides

parallelogram

14. x =

5

26

x=

rhombus

equal sides and equal angles

square

parallelogram with equal sides

3 7

5. 4 triangles

15. 0.5, 1.5, 2.0

6. 6 triangles

16. k = –2 17. 6 min 18. 10 candies 19. 2, 4,

41

20. a) 16, b) 1.5, c) –2, d) any number 7

15

3420°

12, 21 10. 11.

6

3. Polygons and Quadrilaterals (p. 27–32) 1.

It must be made up of line segments only.

2.

8.

9.

19

e) no solution, f) 4.75, g) 4.5, h)

7.

12.

M = 136°, K = 44° 13. A, D 14. 120 cm²

166

4. Bar Graphs and Histograms (p. 33–52)

11.

Number of products

Discount, %

1-7

1

8-15

2

16-30

3

31-45

5

46-60

7

61-75

10

76-90

15

91+

20

1. A. 1 B.10 C. 21 D. 50 2. А. 5 B. 19 C. 36 D. 199 3. A. 5, B. 9, C. 5, D. 4, E. 25, F. 20 4. А. 4 B. 37.5 C.60.5 D.79 5.

Class intervals

55-59

60-64

65-69

70-74

75-79

80-84

85-89

90-94

95-99

Tally marks

|||| |

|

|||

|||

|||| ||||

||

||||

|||

12. 3 rides

Frequency

6

1

3

3

5

2

4

3

13. 4

5

6. A. 43 B. 43 C. 46

14. 17

7. The range: 19

15. 28 m

10

13

16

18

21

24

25

26

27

29

4

1

2

2

2

1

1

2

4

1

16. seal, 6 hours 17. March 235 18.

8.

10

14

15

16

17

2

3

1

2

1

Lifespan (in years) 30 20 10

The range: 7 9. A Chocolate

Rabbit 30-34

35-39

40-44

45-49

50-54

55-59

60-64

65-70

0

0

0

3

1

2

1

0

1

1

0

0

0

0

1

0

0

0

2

2

3

Milk Dark

1

White

0

Nut

0

B.

Milk: 17

Dark: 16

White: 13

4 2

0

2 0

2 0

0

C.

D.

Least sold: Dark

Dark ≈ 38.71

Most sold: Nut;

Nut: 12.

Milk ≈ 52.29;

White ≈ 43.14; Nut ≈ 63.14

Chocolate with nuts has the highest average sales.

10. Diamond

167

5% 76

19.

40

Cat

Horse

Lion

Animal

Crop harvested (in tonnes)

32 24 16 8 Day 1

20. 54 mm 21. 71 m

Day 2

Day 3

Day 4

Days

22.

140

25.

Number of phones sold

400

120 100

300

80 60

200

40 20

100 Mon

23.

15

Tue

Wed

Days

Thu

Fiction Mystery Comics Science Book genres

26. A. False B. False

Number of plants

27. A. True B. False C. False

12

28.

9 6

12

3

8

Spring

24.

60

Number of books sold

Summer

Autumn

Seasons

Length (m)

4

Beltfish

Apples harvested (kg)

Whale shark

Blue Marlin Fish species

29.

Height of the tallest towers (m)

50 40

900

30

750

20

600

10

450

Day 1

Day 1

Day 1

Days

300 150

Tokyo

Dubai

Moscow

Cities

30. A. 230 cm B.190 cm C. Athlete1: 200 cm, Athlete 2: 240 cm.

168

34.

31.

Month

Adult

Child

January

1200

400

February

1400

500

March

1600

600

April

1800

700

12

Number of students

10 8 6 4

32.

2 Day

Adult

Child

Total

Monday

120

40

160

Tuesday

140

50

190

Wednesday

160

60

220

Thursday

180

70

250

Friday

180

70

250

A. Thursday and Friday

D. 90 books

В. 1070 books

E. Monday D. 152 million

B. 38 million

E. 35.75 million

C. 134 million

F. 1 : 1

Year

Sale in Q1 (million ₹)

Sale in Q2 (million ₹)

Sale in Q3 (million ₹)

Sale in Q4 (million ₹)

2023

25

37

29

43

2024

31

40

32

49

A. 9 B. 64−69 kg

Number of books read

35.

Number of students

8 7 6 5

C. Thursday and Friday 33. A. 33.5 million

5-10 10-15 15-20 20-25 25-30

4 3 2 1 15-20 20-25 25-30 30-35 35-40 Time interval

(seconds)

36.

D. True

H. True

F. True

J. False

A. False

E. True

C. True

G. True

B. True

I. False

K. False

C. 45–49 kg and 75–81 kg D. 64−69 kg E. 45−49 kg and 81−86 kg, 49−52 kg and 75−81 kg

5. Pie charts (p. 53–60)

F. 120

1. Predators

5. 15

3. Borneo

7. 40

G. 45 H. 7

2. May

4. False

169

6. 25 8. 52

9.

3.

10.

83

0 6–7

11.

–4×25 0²

(–5)² 12⁴

4.

12.

Positive Positive Zero Non-negative 5. –100, –64 6. 5 m, 25 m²

6. Probability (p. 61–64)

7. 289, 1681, –1849,

1. I, C, I, C, I

8. B, C

2. A red ball will be drawn when taking a ball from a bag containing red and blue balls.

When rolling two fair dice, the first die shows 1 and the second die shows 6.

9.

7. 0.5

1 3.

6 1 1 4. , , 0.43 7 5 1 5.

11

2

3

0

5

–1

4

6

0

10

–2

4

9

0

25

1

8. 0.93

10. 6396, 2499, 3591, 8084, 875

9. 200, 56, 0.28

11. 8,8

10. 0.2, 10

6. 0.8

12. 11,11

11. a) 0.6, b) 36

13. c = 16, b = 9, a = 81 14. 4, 5, 8,

7. Squares and Square Roots (p. 65–72)

15. 1.5

2 ,7 5

16. 13

1. x

–4

–3

–2

–1

0

1

2

3

4

17. 0.09

x²

16

9

4

1

0

1

4

9

16

18. 2.25, 5.8, 9.6

2.

Zero: 0², 0

Positive: –3+5, –(–16), (–3)² Negative: –4

49 144 , 64 169

19.

5

20.

19

21. 13 m

170

8. Cubes and Cube Roots (p. 73–88) 1. 2

11. 4800 12.

A.

2. а) 27 cm³, 125 cm³, 64 cm³, the second box is the most spacious. b) 343 cm³, 64 cm³, 512 cm³, 216 cm³, the third box is the most spacious 3.

A. Length 1 cm, B. Width 1 cm, C. Height 1 cm.

D. 1-24, 2-48, 3-72, 4-96, 5-120, 6-144

B.

C.

4. 64 5. A. 1³ + 2³ + 3³ + 4³ + 5³ = (1 + 8 + 27 + 64 + + 125 = 225 = 15² В. 1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (1 + 8 + 27 + 64 + + 125 + 216) = 441 = 21²

D.

C. 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ = (1 + 8 + 27 + + 64 + 125 + 216 + 343) = 784 = 28² 6. А. 3571

В. 9919

C. 25117

7. А. –729; −4913; −125; −1000000.

–8 1 ; 1 ; 512 ; В.

27 8000 3375 27 C. 4.096; 0.015625, 2.863288; 3511.808.

8. А. 12673 is not a perfect cube; 42875 perfect cube; 8778 is not a perfect cube В. 1009875 is not a perfect cube; 238328 perfect cube; 343 is not a perfect cube 9.

A. Multiply by 2; cube root of 216 = 6 B. Multiply by 4; cube root of 1,000 = 10 10.

A. Divide by 2; cube root of 512 = 8 B. Divide by 12; cube root of 216 = 6

171

13. A. 3; B. 5; C. 10; D. 7 14. A 4, B 3, C 1, D 2.

15.

A. 4; 16; 7; 11 6 7 ; ; ; 12 41 18 1 1 1 C. ; ; .

5 10 20

B.

16.

A. –3; –0.3; –21; B. –24; –110.

17.

18.

A. 5832

B. 4096

C. 13824

5832 = 2³ × 3⁶ = 2³ × 9³

4096 = 2¹²

13824 = 2⁹ × 3³ = 8³ × 3³

³ 5832 = ³ (2×9)³ = 2 × 9 = 18

³ 4096 = ³ 212 = 24 = 16

³ 13824 = ³ (8 × 3)2 = 8 × 3 = 24

2

5832

2

4096

729

2

2

2048

2 3

81

2

512

2

2916

3

243

3 3

3

2

27

2 2

9

2

3

2

1

2

2 2

1024 256

2 2

3456

2

864

2

2

32

2

16

2 2

8

3

2

3

4

1

6912

2

128 64

13824

3

1728 432

216

108 54 27 9 3

1

19. A. 4, B. 8, C. 15.

20. A. 259, 21.

B. 6, 5.

C. 87, 17.

21. A. 141, 31.

B. 242 , 23

D. 200, 10.

C. 359, 19

D. 465 , 17.

22. Yes. 23. Yes. 24. 16

172

9. Percentage and Its Applications (p. 89–100) 1. A. ₹80,000 2. A. ₹544

B. ₹1,30,000

B. ₹257

C. ₹2,20,000

C. ₹101

3. A. ₹3,13,720 B. ₹1,260,000 C. ₹1,920,000 D. ₹16,05,000 C. ₹27,907

1. A. Simple interest ₹33, Amount ₹583.

B. Rate 18%, Amount ₹1,290.

C. Time 2 years, Amount ₹1,830.

D. Principal ₹10,472.73, Amount ₹11,192.73.

2. Matthew should choose deposit B.

4. A. ₹12,903

B. ₹16,216

5. А. 2

В. 4

C. 3

D. 1

3. A. ₹1,125

6. A. 3

B. 4

C. 2

D. 1

B. ₹16,125

7. Cost price of first laptop: ₹1,20,000 Cost price of second laptop: ₹60,000 Total profit percentage: 5% 8. 20.32% 9. ₹3,21,000 10. 60% 11. A. Discount: ₹36,000 SP: ₹2,04,000 B. Discount: = ₹19,200 SP =₹2,20,800

4. A. ₹77.33 (84 days) B. ₹139.92 (152 days) 5. Amount: ₹1,29,113.41 Compound Interest: ₹30,613.41 6. 4.81% 7. Option B earns ₹137.66 more.

12. A. SP ₹17,060; B. ₹15,350

8.

13. A. 16.78%; B. 10.91% 14. ₹8,000

a) A (₹2,38,203.20) > B (₹2,37,422.70) > C (₹2,37,011.92) > D (₹2,36,494.89)

15. Regular price: ₹4500. Discount: 73.33%

b) Option A is the most profitable.

16. ₹68,880

9. C

17. ₹49,421 18. ₹8,685 19. 8% 20. A. ₹1,47,560; B. ₹1,43,607.5 21. ₹ 3200 22. ₹2,32,200 23. 25% 24. Store 1 ₹208,680 Store 2: ₹200,040, Store 2 is more profitable. (cheaper by ₹8,640)

173

10. Compound Interest (p. 101–106)

10. C 11. C 12. Both A and R are true, and R is the correct explanation of A.

11. Algebraic Expressions (p. 107–118)

1.

Expressions

Terms

Coefficient of x

Factors

Constant

A

-

B

2

C

–9

D

–7

2.

A. –x² + 2x + 3 В. –2x² + 3x + 4 C. –3x² + 2x D. 6x + 12 3.

4.

1. 10ab + 18, 4ab + 4; 2. 10ab, 6ab; 3. 5ab² + 8, 7ab² + 7; 4. 7a²b – 9, 10a²b + 18; 5. 10b + 18y + 9, 7b + 3y – 5; 6. 2ab² – 8b + 3, 10ab² – 7b – 18.

Monomials:

5.

2pqr;

A. (m−n)²−3mn

w8x²y² Binomials: 10ab + 18; xy + 5xc; xy²– 10; ab + abc²

B. x²+3y+xy C. 3mn−4m+n D. (p+q)/3(p+q) E. (p+q)(p−q) F. mn/3mn 6.

Trinomials:

A. 12x + y + 9.

7c – 9ab – 90;

B. 3x² + 13x + y + 9

abc – 6a +5; a²bc – a +7 Quadrinomials: a2 + 5abc – za² – 10; a² + 5a2b² + 6b² – 6c; 2ab + abc – 3c – 4

C. 2x + 14y − 19 D. 3x² − 3x + 10y − 13 7.

A. 6x + 7 B. 14x + 8y − 5 C. −x2 + 5y − 8 D. −7x + 3y + 16 E. −4x² − 2y − 18

174

8.

A. −y − 46 B. 46n + 3m C. −4y² + y + 6 D. 3m² + 19m − 46 9.

A. 2y² − 5y + 12 B. 2y² − 2y − 2 C. −y² + 8y − 10 D. 3y² − 2y − 6 10. a) 3xy + 3x b) 35xy + 7y c) −xy − 4x + 7y + 28 d) 18xy + 9x 11.

A. 24xy + 9y − 32x − 12 В. 4a²b − 2a² + 8b² + 16b − 10 C. z⁴ − z³ + 4z² + 20z

В. 6,56,25,00,000 C. 50,70,00,00,000 15.

A. V = (3a + 2b)(a² − b + 5)(4ab) = 12a⁴b + + 8a³b² − 12a²b² + 60a²b − 8ab³ + 40ab² m³ B. 34,020 m³ C. 9500 m³ D. 24,520 m³

12. Area of polygons (p. 119–124) 1. 131.93 cm² 2. 103.25 cm² 3. 80 cm² 4. 30.96 cm² 5. 27 unit², 36 unit²

D. 9p⁴ − 55p³ − 59p² + 21p

6. 12 unit², 11.5 unit²

12. A. Area = 8x³ + 14x² − 43x + 21; When x = 2, Area = 55

7. No, yes, yes 8. –, 1100 m², 800 m²

B. Area = 9y³ − 6y² + 19y + 18; When y = 1, Area = 40

9. Shape A: ~357 cm²; Shape B: ~2153 cm²; Shape C: ~66 cm²; Shape D: ~1314 cm²; Shape E: ~457 cm²; Shape F: ~479 cm²; Shape G: ~2072 cm²; Shape H: ~94 cm²; Shape I: ~344 cm²; Shape J: ~14 cm²

C. Area = 5z⁴ + 20z³ + 14z² − 3z; When z = −1, Area = 2 D. Area = 20m⁴ + 28m³ + 27m² + 11m + 28; When m = 0, Area = 28 13. A. 42a² − 16ab + 9b² + 30ac − 4bc B. −a²c − 30a²bc + ab³ + 4ab² − 40abc + 11b²c– − 10bc − 2c C. a³ − 6a² − 2ab² + 12b² + 9ac + 18abc − 5a − – 54c − 9bc + 30.

D. −6a³ − 9b³ + 20ab − 15ac + 6bc

175

14. А. Volume = (x + y + z)(2x − y + 3z)(5xy) = = 10x³y + 5x²y² + 25x²yz − 5xy³ + 10xy²z + + 15xyz²

13. Surface Area and Volume of Solids (p.125–128) 1. 108 cm²

3. 5,657 cm²

6. 6 m²

2. A. 62.86 cm² B. 88 cm²

4. Yes

7. 8 cm

5. 25 cm²

8. 40 cm

14. Exponents and Powers (p. 129–136)

10.

A: 5.6 × 10⁷ < 5.7 × 10⁸ B: 9.5 × 10⁶ < 8.9 × 10⁸

1.

C: 3.2 × 10⁸ < 7.0 × 10⁸

A. 2⁶>6²

11.

B. 5⁹>9⁵

A.

C. 4⁷>7⁴ 2.

1) A. 2 × 10² + 3 × 10¹ + 4 × 10⁰ + 5 × 10⁻¹ + 6 × × 10⁻² B. 7 × 10³ + 0 × 10² + 8 × 10¹ + 9 × 10⁰ + 0 × × 10⁻¹ + 0 × 10⁻² + 4 × 10⁻³ C. 1 × 10⁻¹ + 2 × 10⁻² + 3 × 10⁻³ + 4× × 10⁻⁴ + 5 × × 10⁻⁵

B.

D. 6 × 10³ + 0 × 10² + 0 × 10¹ + 0 × 10⁰ + 0 × × 10⁻¹ + 0 × 10⁻² + 7 × 10⁻³ + 8 × 10⁻⁴ E. 9 × 10³ + 8 × 10² + 7 × 10¹ + 6 × 10⁰ + 5 × × 10⁻¹ + 4 × 10⁻² + 3 × 10⁻³+ + 2 × 10⁻⁴ + 1 × 10 2) A. 20

B. 28

C. 15

D. 21

E. 45

3) E (9 numbers) 4) B, D or E 3. a) 5⁻²; b) 10⁻¹; c) x⁻⁶; d) a⁻¹; e) 23⁻⁴ 4. a) 5. a)

1 1 1 1 1 ; b) ; c) ; d) ; e) 81a4 343 2 y10 b 1 1 1 ; b) ; c) – 9 25 6

25 27 6. a) 3; b) ; c) 10000; d) 4 343 7. a) 8.

1 4

11 17 ; b) ; c) 453; d) –7 72 12

12.

13. A. 3, B. 2, 2.25

27 8

C. 6, 64 14. b = 4; c = 2 15.

37, 64 0.5, –25

9. 81

176

15. Direct and Inverse Proportions (p. 137–144) 1. 6:3

137 times heavier than 17. The panda is 15 koala.

18.

= 2, 2

2. 6:3 =

16. 20:139

a) Yes

3. 4:6 or 2:3

b) 2

4. 7:4

7 3 should be ; 1.75 5. Mistake: 4 4 6.

c) Row x: 6, 14 Row y: 10, 40 19. a) Yes b) k = 120 c) Row p: 6, 4 Row q: 12 20. False, True,

7.

21.

1. B.

6 7

2. C.

x

33

11

132

176

55

8.

y

12

4

48

64

20

Case1: 3:1

22. 7680 words

Case2: 2:1 9. 5:2,

23. 126 km

5 2

7 2 11. 300:60

24. 10 min 25. 9 hours 36 minutes

10. 7:2,

12. 300:60 = 13. 300:100

5 = 5; 5 1

16. Factorisation and Division of Algebraic Expressions (p. 145–152)

3 = 3; 3 14. 300:100 = 1 15.

1:4

1. 6, 8, 15 2. 4x, 2x, 2x 3. (5p + 1)(p − 1)

2:3

177

(2m – 3)(2mn – 3n + 2m – 4) (x + 6)(3x − 11)

4. 1 column:

2 column:

17. Linear Graphs (p. 153–162) Water level (m)

1.

2.8

5.

2.6 2.4 2.2 2

6.

1726 1752 1898 1977 1986

7.

2.

Year

Height (m) 1.3

1.25 1.2

1.15 1.1

1.05

8. 25, 1, 16

1

9. 40 10. 31 11. 9

4

3.

12. 6x, 6x; 5x, 5x; 4x, 4x 14.

15.

4x³y²

16.

1

4.

Age (years)

5

9

13

17

Age (years)

24 18

2z⁴ – 3z² + 4

12

5m²n + 3m – 2n²

6

3p² – 2pqr + r²

8

Rainfall (mm)

2x² – 3x + 5

2a²bc – 3ac² + 4b²

7

Sleep (hours)

–5a²b³c –4p³q²

6

16 14 12 10 8 6

13. 120, 180, 160

–4m³n²

5

June July Aug. Sept. Oct.

Month

178

5. ₹600 6. 75 cm 7. 50 minutes 8. 9 m 9. А. False, B. True, C. True 10. 27°C 11. 10 minutes 12. 3; 4; 5; 7 13. 3000 million 14. 100 km 15. А. True B. False 16. А. True B. False 17. А. True; B. True 18. A. True; B. False 19. A. True; B. False 20. 70 m

Answers (pages 160 - 161) 179