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Acknowledgements Academic Authors: Muskan Panjwani, Evgenii Todorov Creative Directors: Alena Sizintseva

Book Production: Larisa Pavlenko, 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 7 ISBN: 978-93-89789-96-6 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.

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.

Open the Uolo Learn app.

Tap on DinoLab.

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.

Open the Uolo Learn app.

Click on QR icon in the upper right corner.

Scan QR code next to the task.

n Sca

cod

Curricular Programs

iii

Contents 1. Integers Addition and Subtraction of Integers �� 1 Multiplication and Division of Integers ��6 2. Fractions and Decimals Operations on Fractions �� 14 Operations on Decimals ��27 3. Data Handling Mean, Median and Mode �� 33 Organisation and Representation of Data �� 43 4. Introduction to Probability Understanding Probability �� 50 5. Simple Equations Linear Equations in One Variable �� 57 6. Angle Pairs and Parallel Lines Angle Pairs �� 75 Parallel Lines ��95 7. Triangles and Their Properties Understanding Triangles and Their Properties �� 104 Right-angled Triangles �� 106 Congruence �� 118 8. Ratio and Proportion Ratio and Proportion �� 123

9. Percentage, Profit and Loss, Simple Interest Percentage and Its Applications �� 141 Profit and Loss �� 151 Simple Interest �� 156 10. Rational Numbers Introducing Rational Numbers �� 157 Operations on Rational Numbers �� 164 11. Construction of Triangles Constructing Triangles �� 169 12. Perimeter and Area Squares, Rectangles, Triangles and Parallelograms �� 175 Circles �� 186 Word Problems on Perimeter and Area �� 187 13. Algebraic Expressions Generating Rules in Formulas and Patterns �� 189 Algebraic Expressions and Terms �� 191 Addition and Subtraction of Algebraic Expressions �� 194 14. Exponents and Powers Exponents �� 201 15. Symmetry Understanding Symmetry �� 210 16. Visualising Solid Shapes 3D-Shapes and 2D-Shapes �� 215 Answers �� 219

1. INTEGERS ADDITION AND SUBTRACTION OF INTEGERS 1. Write these numbers in the ascending order. Then find the sum of the smallest and largest numbers. −12, 7, 0, −3, 5, −8

2. Find whether each statement is true or false. Write T for true and F for false. a) −5 > −11

c) 0 is less than every positive integer.

b) |−6| > |4|

d) −1 < 0

3. Compare the numbers in each pair. Write <, = or > in the blanks. a) −3 b) | − 7 |

2 | − 4|

c) −10

− 15

d) 0

|−5|

4. Look at the number line given below. Mark the following integers on it: −5, −3, 2, 4, 5 0

Integers

5. Which integers do the letters on the number line show? Join each letter to the correct integer.

−2

−5

6. Look at the number line given below. Use it to complete the following sentences. 0

a) The distance between 0 and 3 is

b) The distance between 5 and 2 is

. .

c) The distance between −5 and −1 is d) The distance between −3 and 4 is

e) The distance between 3 and −2 is

7. a) On the given number line, mark a point that is halfway between −4 and 6. 0

1 Integers

b) On the given number line, mark all the integers that lie between −3 and 3. 0

c) On the given number line, mark the midpoint between −5 and 1. 0

8. Find the value of each of the following. Write your answers in the blanks. a) |2| =

c) |4−2| =

b) |− 9| =

d) |7−9| =

e) |–5 + 8| =

9. Solve each of the following for the integer x. Write all the possible values of x. a) |x| = 2

b) |x| = 0

Integers

10. Complete the following sentences. a) The distance between −4 and 5 is

. .

b) The distance between −7 and −1 is

11. Add. a) −12 + 7 =

с) −3 + (−8) =

b) 5 + (−9) =

d) 14 + (−14) =

12. Add. −5 + 3 + (−7) + 2 + 10

13. Find the missing number in each equation. Write it in the box. a) −6 +

+ (−4) = −10

c) 7 +

= −5

14. Subtract. a) 9 − (−14) =

b) 0 − (−25) =

c) −6 − (−6) =

Integers

15. Rewrite each subtraction statement as an addition statement. Then simplify it. a) 5 − (−3)

b) −10 − 8

16. Find the missing number in each of the following equations. Write it in the given box. a)

− 4 = −9

b) 6 –

= 11

− (−7) = 2

17. The temperature at 6 a.m. was −3 ℃. By noon, it increased by 9 ℃. What was the temperature at noon?

18. A submarine is at a depth of 450 metres below sea level. It rises 120 metres, then dives 200 metres. What is its final position from the sea level?

Integers

19. Ravi’s bank account has a balance of ₹1,500. He withdraws ₹1,200. What is his balance now?

20. In a game, a player gains or loses points as follows: • +8 points for each correct answer • −5 points for each incorrect answer A player answered 12 questions and scored 5 points in total. How many questions did the player answer correctly?

MULTIPLICATION AND DIVISION OF INTEGERS 21. Find the product. a) (−12) × 8 =

с) 15 × (−4) =

b) (−7) × (−6) =

d) (−5) × 0 =

Integers

22. Find the missing number in each equation. Write it in the box. a) (−3) ×

= 21

× (−8) = −64

c) (−9) ×

23. A car manufacturer finds that in each trial a car travels 2 km less than the expected distance. How much less distance does it travel over 5 trials?

24. The product of four integers is positive. Exactly three of these integers are negative. Is this possible? Explain your reasoning.

25. Solve each division problem. How many of the quotients are integers? Tick (✓) them.

a) 75 ÷ (−8)

c) 48 ÷ 9

b) −56 ÷ 7

d) 39 ÷ (−3)

Integers

e) −81 ÷ 9

26. Divide. Write the quotient in the box. a) 84 ÷ (−6) =

с) −100 ÷ (−25) =

b) −96 ÷ 12 =

d) −45 ÷ 9 =

27. Find the missing number in each equation. Write it in the box. a)

÷ (−5) = −9

b) −48 ÷

÷4=0

28. Raman is exploring an underwater cave. First, he swims downward at 15 metres per minute for 6 minutes. Then he swims upward at 12 metres per minute for 8 minutes. Finally, he swims downward at 10 metres per minute for 5 minutes. What is his final position from the starting point?

Integers

29. A lift descends 4 floors every 8 seconds. How many floors does it descend in 1 minute?

30. A trader earns ₹250 for each sale but loses ₹180 for each refund. He makes 6 sales and 3 refunds. What is his total profit or loss?

31. Look at the expression. Which property of operations does it show? Tick (✓) your answer. (−5) + (−7) = (−7) + (−5)

a) Commutative property of addition b) Associative property of addition c) Distributive property

32. Look at the expression. Which property of operations does it show? Tick (✓) your answer. (−3) × (12 + 5) = (−3) × 12 + (−3) × 5

a) Associative property of multiplication b) Distributive property c) Commutative property of multiplication

Integers

33. Look at each expression. Which property of operations does it show? Tick (✓) your answer. a) a + (b + c) = (a + b) + c

c) a × (b + c) = a × b + a × c

Associative property

Commutative property

Distributive property

b) a × b = b × a Associative property Commutative property Distributive property

34. Rearrange the numbers to make the calculation easier. (−2) + 8 + (−5) + (−1) Explain which properties you used.

35. Simplify using the distributive property. (−4) × (6 − 3)

Integers

36. Fill the blanks. a) The additive inverse of 7 is

. .

b) The additive inverse of −11 is .

c) The additive inverse of 0 is b) The additive inverse of 25 is

37. Find the missing number in each equation. Write it in the box. a) 5 +

+ (−9) = 0

c) 0 +

38. Fill in the blanks. a) The multiplicative inverse of 4 is

. .

b) The multiplicative inverse of −3 is c) The multiplicative inverse of 1 is

b) The multiplicative inverse of −10 is

39. Find whether each statement is true or false. Write T for true and F for false. a) The additive inverse of x is always −x. b) The number 0 has a multiplicative inverse. c) The multiplicative inverse of −1 is −1.

Integers

40. Simplify. a) (−12) + 15 × (−2) ÷ 3

b) (6 − (−4)) × (−3)

c) −52 + 3 × (−4)

41. Look at the equation. Insert parentheses in the correct places to make it true. −8 + 6 × −2 = 4 42. Simplify: (−3) × ((−2) + 5) − 6 ÷ (−3)

43. Find the missing number in the following equation. Write it in the given box. + 12 × (−3) = 0 44. A student wrote: −8 − (−5) = −8 − 5 = −13. Find the mistake. Write the correct answer.

Integers

45. Start at 0 on a number line. Follow these steps: +4, −7, +5, −3, +6. Where do you end up? Write the final number.

46. Complete the given magic square. Use integers from −4 to 4 exactly once. Each row, column and diagonal must add up to 0. –4

47. The sum of three consecutive integers is −9. Find the integers.

Integers

2. FRACTIONS AND DECIMALS OPERATIONS ON FRACTIONS 1. Look at the shaded portion of each figure. Write a fraction for it in the simplest form.

2. Read each statement. Shade as instructed.

a) Shade

3 4

of the rectangle.

b) Shade

2 5

of the circle.

c) Shade

5 8

of the square.

Fractions and Decimals

3. Read each sentence. Write it as a fraction in the simplest form. a) A bar is divided into 6 equal parts. 4 of them are shaded. b) A pizza is cut into 10 equal slices. 7 of them are eaten. c) A class has a total of 20 students. 15 are wearing glasses.

4. Write each improper fraction as a mixed fraction.

45 10

41 8

48 18

17 5

91 12

5. Write each mixed fraction as an improper fraction.

a) 3

d) 2

4 5 2 3

19 5

Fractions and Decimals

b) 7

e) 1

1 6 3 4

с) 5

f) 4

3 8 7

6. Solve.

a) Find a such that 2

7 9

b) Find a and b such that

c) Find m such that m +

3 8

58 15

=a+

83 8

7. Reduce each fraction to its simplest form.

12 18 28 42 63 84

2 3

45 60 40 50

с)

120 150

24 55 81 81

108

Fractions and Decimals

8. Convert each fraction to an equivalent fraction with the given numerator or denominator. a) d)

3 5 9

= =

20 3

12 18

= =

6 2

7 11

= =

21 24 33 45

9. Find the fractions that are equivalent. Join them.

3 9

1 3

4 8

3 6

Fractions and Decimals

10. Write each fraction as an equivalent fraction with denominator 60. a)

1 2 7

2 3 9

3 4 11 15

11. Rewrite each pair of fractions as equivalent fractions with least common denominator. a)

3 4 5 8

and

6 3

and

2 3 7 9

and

12 11

and

12. Rewrite the fractions in each group as equivalent fractions with least common denominator. a)

1 2 4 7

2 3 5 9

3 4 6

3 5

5 8

Fractions and Decimals

13. Compare the fractions. Write <, >, or = in the boxes. a) d)

5 8

14. Compare the fractions. Write <, >, or = in the boxes. a) d)

b) e)

3 5

15. Arrange the fractions in increasing order.

1 2 4 7

3 8 5 9

2 3 7

5 6 3 5

Fractions and Decimals

16. Solve. Write your answers in the simplest form. a)

3 7 6

2 7 3

5 9 8

–

4 9 2

17. Solve. Write your answers in the simplest form. a)

3 4 5 8

5 6 3

2 3 7 9

–

12 1

2 18. A tank is full of water. Then more water equal 8 1 3 of the tank's capacity is added. After that, to 4 10 of the total capacity is drained. What fraction of the tank is filled now?

Fractions and Decimals

19. Solve. Write your answers in the simplest form.

3 5

4 9

7 8

–

5 6

3 7

–

1 6

Fractions and Decimals

20. Solve. Write your answers in the simplest form.

2 5

7 8

–

3 8

21. Multiply. Write your answer in the simplest form.

3 5 5 9

4 7 3

7 8 8

11 9

Fractions and Decimals

22. Multiply. Write your answer as a mixed fraction in the simplest form. a) 2

b) 1

c) 3

d) 5

1 3

1 2

3 8

2 3

3 4

×2

2 5

4 7

×1

3 5

Fractions and Decimals

3 cup of sugar to make 600 grams 4 of cookies. How much sugar is needed to make 23. A recipe uses

400 grams of cookies?

24. Divide. Write your answer in the simplest form.

3 5 5 9

4 7 3

7 8 8

11 9

3 metre long. How many pieces can 4 1 you cut from the ribbon if each piece needs to be 8 metre? 25. A ribbon is

Fractions and Decimals

26. Divide. Write your answer in the simplest form.

a) 2

b) 1

c) 3

d) 5

1 3

1 2

3 8

2 3

3 4

÷2

2 5

4 7

÷1

3 5

Fractions and Decimals

27. A book has 240 pages. Sahil read How many pages has he read?

7 of the book. 12

2 28. 9 of a number is 18. Find the number.

1 of its 4 original price. If the original price was ₹32, what is 29. A shop increases the price of a toy by

the new price?

30. Arrange the given numbers in ascending order. a) 0.45, 0.5, 0.405

b) 0.82, 0.805, 0.8

c) 0.125, 0.12, 0.13

Fractions and Decimals

OPERATIONS ON DECIMALS 31. Write each decimal as a fraction in the simplest form. a) 0.5 =

b) 0.25 =

c) 0.125 =

d) 0.75 =

e) 0.6 =

f) 0.2 =

32. Write each fraction as a decimal. a) d)

3 4 11 25

10 2 5

33. Solve. a) 3.5 + 2.4 =

b) 7.8 – 3.2 =

c) 4.75 + 5.25 =

d) 10 – 6.35 =

34. Subtract.

a) 12.8 – 4.65 =

b) 3.15 + 7.45 =

c) 8.08 – 2.39 =

d) 5.25 + 6.75 =

Fractions and Decimals

20 1 8

= =

35. Solve. a) 0.75 + 0.85 + 1.25

b) 5.8 − 2.35 + 4.2

c) 6.05 + 3.95 − 2.1

d) 7.7 − 3.15 − 2.25

e) 15.25 + 7.6 − 12.45

f) 8.08 − 3.5 + 6.92

g) 20 − 7.35 − 4.65

h) 11.11 + 9.99 − 10

Fractions and Decimals

36. Multiply.

a) 3.5 × 2

b) 6.25 × 4

c) 7.5 × 0.8

d) 0.75 × 0.6

e) 5.4 × 1.25

f) 12.5 × 0.08

g) 4.2 × 1.5

h) 3.14 × 2.5

Fractions and Decimals

37. Divide. a) 7.5 ÷ 5 =

b) 8.4 ÷ 4 =

c) 6.6 ÷ 3 =

d) 9.9 ÷ 6 =

e) 18.9 ÷ 3 =

f) 7.35 ÷ 5 =

38. Divide. a) 16.5 ÷ 3

b) 22.4 ÷ 7

c) 19.6 ÷ 4

d) 33.75 ÷ 9

Fractions and Decimals

39. Divide. a) 6.4 ÷ 0.8

b) 7.5 ÷ 1.5

c) 8.25 ÷ 0.5

d) 9.6 ÷ 1.2

e) 3.75 ÷ 0.025

f) 7.92 ÷ 0.04

g) 6.3 ÷ 0.009

Fractions and Decimals

40. A rope is 7.5 metres long. It is cut into 5 equal pieces. How long is each piece?

41. A bag of apples weighs 4.8 kilograms. If each apple weighs 0.24 kilograms, how many apples are there in the bag?

42. A car travels 182.4 km in 3.2 hours. Find its average speed in km/h.

43. A decorative lace costs ₹5.75 per metre. How much will 8.4 metres of decorative lace cost?

Fractions and Decimals

3. DATA HANDLING

MEAN, MEDIAN AND MODE 1. The table shows the weights and heights of some savanna animals. Study the table and fill in the blanks. Animal

Weight (kg)

Height (cm)

Giraffe

1200

600

Elephant

7000

350

Lion

195

120

Zebra

430

140

Leopard

Rhinoceros

4500

200

Ostrich

130

230

a) What is the height of a giraffe?

cm. cm.

b) Choose the animal with the smallest height. Write its height: c) Which animal is 140 cm tall? d) Which animal has the highest weight? e) Choose the animal that weighs 4500 kg. Write its height:

Data Handling

cm.

2. The table shows the area and the maximum depth of the world’s oceans. Study the table and fill in the blanks. Ocean

Area (million km2)

Maximum depth (m)

Pacific

169

11,022

Atlantic

8,442

Arctic

5,527

Indian

7,725

a) Which ocean has the largest area? b) What is the maximum depth of the Pacific Ocean?

c) Arrange the maximum depths of the oceans in ascending order. m m m m d) Arrange the oceans in descending order of their areas. Which ocean comes first? e) Which ocean has the second largest area?

Data Handling

3. Which of the following is the formula for finding the mean of a given observations? Tick (✓) the correct option. a)

Mean =

Number of observations Sum of all observations Sum of all observations Number of observations

Mean = Number of observations × Sum of observations

Mean =

Sum of all observations 2

4. List five consecutive even numbers starting from 2. Find their sum.

5. List five consecutive even numbers starting from 2. Find their arithmetic mean.

6. The sum of four numbers is 41. Find their arithmetic mean.

Data Handling

7. The mean of ten numbers is 12.2. Find their sum.

8. The mean of two numbers is 30. If one number is 25, what is the other number?

9. The mean of four numbers is 50. If three of the numbers are 40, 41, and 42, what is the fourth number?

10. Find the mean of the following set of numbers. 7, 7, 7, 7, 10, 10

11. In a set of numbers, 10 occurs 2 times, 20 occurs 5 times, and 30 occurs 3 times. Find the mean.

Data Handling

12. Ayesha practiced playing basketball and took 30 shots in total. She made 12 successful two-point shots and 8 successful three-point shots. a) What is the total number of points Ayesha scored?

b) What is the mean number of points per successful shot?

c) What is the mean number of points per shot, including both successful and unsuccessful shots?

13. The mean of 5 numbers is 50. If one more number is added, the mean increases by 3. Find the added number.

14. The mean of 6 numbers is 40. One of the numbers, 35, is removed. Find the new mean of the remaining numbers.

Data Handling

15. The table shows data about football matches at a school. Study the table and answer the questions. Match

Players

Referees

Spectators

136

118

125

149

a) What is the total number of spectators across all four matches?

b) What is the mean number of spectators per match?

c) What is the total number of players across all four matches?

d) What is the mean number of players per match?

e) What is the total number of people (players, referees, and spectators) across all four matches?

f) What is the mean number of people (players, referees, and spectators) per match?

Data Handling

16. Match the terms with their correct definitions. mean

a) The middle value in a set of data when arranged in ascending or descending order.

mode median range

b) The difference between the largest and smallest values in a data set.

c) The sum of all values in a data set divided by the total number of values.

d) The value that appears most frequently in a data set.

17. Find the mode of this data: 2, 3, 3, 3, 4, 4, 6, 6, 8.

18. Find all modes of this data: 2, 2, 3, 5, 5, 7, 8, 8.

19. Arrange the following numbers in increasing order and find their mode. 35, 25, 15, 20, 35, 20, 12, 35, 30

Data Handling

20. Arrange the numbers in increasing order. Then find all the modes of the data set. 35, 25, 15, 20, 30, 20, 12, 35, 30

21. The table shows how many books each of five friends read in three seasons. Study the table and answer the questions. Name

Autumn

Winter

Spring

Ram

Gagan

Shyam

Rakesh

Ashish

a) Find the mode of the number of books read in autumn.

b) Find the mode of the number of books read in winter.

c) Find all modes of the number of books read in spring.

Data Handling

22. Find the median of the following data. 3, 3, 3, 4, 5, 6, 6

23. Find the median of the following data. 35, 25, 15, 20, 35, 20, 12, 35, 30

24. Find the median of the following data. 2, 3, 3, 3, 4, 5, 6, 6

25. Find the median of the following data. 25, 15, 20, 35, 20, 12, 35, 30

Data Handling

26. The median of the given data is 18.5. What is the value of x? 20, 15, 30, x

27. The table shows how many children of different ages were at a party. Study the table and answer the questions. Age (years)

Number of children

a) How many children were 12 years old? b) How many children were at the party in total? c) What is the sum of the ages of all the children? d) What is the mean age of the children at the party? e) Find the mode of the ages of the children at the party. f) Find the median age of the children at the party. g) Find the range of the ages of the children at the party.

Data Handling

ORGANISATION AND REPRESENTATION OF DATA 28. The chart shows how long different animals live. Study the chart and answer the questions.

Lifespan (in years) 30 25 20 15 10 5 0

Rabbit

Hen

Wolf

Dog

Cat

a) How long do cats live? b) How long does a wolf live? c) Which animal lives the longest? d) What is the difference in lifespan between a hen and a dog? e) Which animal has a shorter lifespan, a dog or a rabbit?

Data Handling

Animals

29. The chart shows how many students received each grade in mathematics this year. Look at the chart and answer the questions. Number of students 70 60 50 40 30 20 10 0

Grades

a) How many students scored grade C? b) How many more students scored B than A? c) Which grade was scored by the highest number of students?

Data Handling

30. The chart shows the heights of famous towers around the world. Study the chart and answer the questions. Height (in metres) 100 90 80 70 60 50 40 30 20 10 0

Tower of Pisa

Spassky Tower

Big Ben

Tower of Belem

Tower names

a) What is the height of the Spassky Tower? b) Which tower is the tallest? c) Which tower is the shortest? d) Which tower is about three times taller than the Tower of Belem?

Data Handling

31. The chart shows the amount of rainfall in Mumbai over a period of six months. Study the chart and answer the questions. Rainfall (in mm) 70 60 50 40 30 20 10 0

January

February

March

April

May

June

Months

a) How much rainfall was there in May? b) Which month had the lowest rainfall? c) Which month had more than 60 mm of rainfall? d) What is the total amount of rainfall from March to May?

Data Handling

32. The chart shows the speed of different animals. The cheetah can run at 150 km/h. Draw a bar to show its speed and colour it.

Speed (in km/h) 350 300 350 200 150 100 50 0

Falcon

10 11 1 2

8 7

Data Handling

Gazelle

Ostrich

Cheetah

Animals

33. The children measured the length of insects in their biology lesson and drew a chart. A beetle is 2.1 cm long and an ant is 0.6 cm long. Draw and colour bars to show their lengths.

Length (in cm) 2.5 2 1.5 1 0.5 0

Mosquito

Beetle

Bee

Ant

Insects

Data Handling

34. The table shows how many gadgets were sold at a store over four days. Use it to complete the chart. Number of gadgets sold Gadgets

Mon

Tue

Wed

Thu

Tablets

Phones

Laptops

Total number of gadgets sold 140 120 100 80 60 40 20 0

Data Handling

Mon

Tue

Wed

Thu

Days

4. INTRODUCTION TO PROBABILITY

UNDERSTANDING PROBABILITY 1. What are the chances of the given events happening? Use sure, impossible, unlikely, likely, or equally likely to fill each blank. a) A rock talking in human language. b) The sun setting in the evening. c) Tossing a coin and getting tails. d) A randomly picked clover turning out to be a four-leaf clover.

e) Finding your notebook where you left it.

2. A fair die is tossed. Write the chance of each event happening. Use sure, impossible, unlikely, likely, or equally likely to fill the blanks. a) Getting a number less than two. b) Getting a number greater than six. c) Getting a number greater than three. d) Getting a number greater than two. e) Getting a number less than seven.

Introduction to Probability

3. Look the image of the aquarium shown. A fish from it is picked at random. Find the chance of following events happening. Use sure, impossible, unlikely, likely, or equally likely to fill the blanks. a) The fish is green. b) The fish is not green. c) The fish is black. d) The fish is not black. e) The fish is blue or yellow.

Introduction to Probability

4. Rakesh has a collection of 20 crystals. They are shown in the picture. Fill in the blanks.

Fill in the numbers. out of 20.

a) The chance of selecting red crystal is b) The probability of selecting a red crystal is:

c) The probability of selecting a yellow crystal is:

d) The probability of selecting a blue crystal is:

Introduction to Probability

5. A letter is picked at random from a word DINOLAB. What is the probability of picking a vowel?

6. Disha baked 3 strawberry cupcakes, 4 blueberry cupcakes and 5 vanilla cupcakes for her birthday party. Use this information to fill in the blanks. A guest selects a cupcake at random. What is the probability that he selects: a) a strawberry cupcake?

= b) a blueberry cupcake?

= c) a vanilla cupcake?

Introduction to Probability

7. There are 10 keys on the table. Of these, 3 are golden keys, 4 are silver keys and 3 are bronze keys. Use this information to fill in the blanks. You pick a key at random. What is the probability that the key is not made of silver?

8. Malati has a box of colourful geometric shapes for her maths activity. In the box, there are 6 triangles, 5 squares and 11 circles. Use this information to fill in the blanks. She picks a shape at random. What is the probability that it is a triangle or a square?

Introduction to Probability

9. A dice was rolled 100 times. The table shows the results. Use this information to fill in the blanks. Number Rolled

Frequency

a) The experimental probability of rolling 1 is:

100

b) The theoretical probability of rolling 1 is:

c) The experimental probability of rolling an odd number is:

100 d) The theoretical probability of rolling an odd number is:

Introduction to Probability

10. A box has 5 red marbles, 7 green marbles and 8 blue marbles. Shyam randomly drew one marble from the box 50 times, replacing it each time. The table shows the results. Fill in the blanks. Colour of marble

Red

Green

Blue

Frequency

a) The experimental probability of drawing a red marble is:

b) The theoretical probability of drawing a red marble is:

c) The experimental probability of drawing a blue marble is:

50 d) The theoretical probability of drawing a blue marble is:

Introduction to Probability

5. SIMPLE EQUATIONS

LINEAR EQUATIONS IN ONE VARIABLE 1. Match each equation to its solution. a)

x+5=9

x=6

x–7=3

x=4

2x – 2 = 10

x = 10

3x + 4 = 16

x = 20

5x − 2 = 13

x=4

x+2=7

x=3

4x – 3 = 2x + 7

x=5

2 5

x+2=

1 2

x+5

2 (x – 3) = x + 5

Simple Equations

x = 11 x = –30

2. Match each sentence to the correct equation.

Twice a number decreased by 4 is 10.

x + 7 = 15

Seven more than a number is 15.

3x + 8 = 20

Half of a number is equal to 6.

2x − 4 = 10

Three times a number increased by 8 equals 20.

One third of a number increased by 5 is 11. Three quarters of a number minus 2 equals 7. Five sixths of a number plus 4 is equal to 9. Seven tenths of a number decreased by 3 equals 8.

1 2

5 6 3 4 7

10 1 3

x=6

x+4=9 x–2=7 x–3=8 x + 5 = 11

Twice a number minus one-fifth of that number is equal to 200.

Three-fourths of a number plus 12 equals 500.

2x –

Half of a number decreased by 25 is equal to 150. Seven-tenths of a number added to 150 equals 850.

10 3 4

x – 25 = 150 1 5

x = 200

x + 150 = 850 x + 12 = 500

Simple Equations

3. Solve the equations. a) x – 8 = –5 1) x = –5 + 8

b) x +

c) x –

d) x +

2 3

5 4

3 5

7 3

+2=

–

2) x = 3

=2–

Simple Equations

1 2

4. Solve the equations. a) 3x − 5 = 13 1) 3x = 13 + 5

2) 3x = 18

divide both sides by 3

3) x = 6

b) 2x = 10

c) 4x + 7 = −9

d) 5x − 18 = −3

Simple Equations

5. Solve the equations. a) 2x – 1 = 2 1) 2x = 2 + 1 3) x = 3 2

b) 3x + 1 = 5

c) 4x – 1 = 5

d) 5x + 2 = 8

Simple Equations

2) x=3 divide both sides by 2

6. Solve the equations. a) –2x + 4 = –1 1) –2x= –1 –4

2) 2x = –5 change the sign

3) 2x = 5 divide both sides by 2

4) x = 5 2

b) –3x – 5 = 1

c) –4x + 2 = 5

d) –5x – 3 = –1

Simple Equations

7. Solve the equations. a)

3 2

x+3=7

1) 3 x =7 – 3 2 2) 3 x = 4 2

multiply both sides by 2

3) 3x = 8

divide both sides by 3

4) x = 8 3 b)

2 3

3 4

x–1=2

x+2=

5 2

Simple Equations

5 6

x–2=

1 2

8. Solve the equations. a) 2(x + 3) + 2 = 10 1) 2(x+3)=10-2 2) 2(x + 3) = 8

divide both sides by 2

3) x + 3 = 4 4) x = 4 – 3 5) x = 1

Simple Equations

b) 3(x – 5) – 5 = 6

c) 4(x + 2) – 3= 18

d) 5(x – 3) + 15 = 2

Simple Equations

9. Solve for x. a) 2x + 3 = x + 7 1) gather x on one side: 2x + 3 – x = 7 2) x + 3 = 7 3) x = 7 – 3 4) x = 4

b) 5x – 4 = 3x + 10

c) 7x + 2 = 5x – 8

Simple Equations

d) 4x – 7 = 2x + 5

10. Solve for x. a)

3 4

1) use the cross-multiplication property: x × 4 = 3 × 5 2) 4x = 15 3) x = 15 4

7 x

14 5

Simple Equations

divide both sides by 4

x+2 3

x–1

11. Solve for x. a)

1 2

x+3=

1 4

x+5

Simple Equations

b) –3(x – 2) = 2x + 5

2 3

x–4=–

1 6

x+2

d) –4(x + 1) + 3 =

Simple Equations

1 2

x–5

12. A shop sells mangoes at ₹50 per kilogram. Priya bought some mangoes for ₹350. Tick (✓) the equation that can be used to find how many kilograms of mangoes she bought. x

a) 50 + x = 350

b) 50x = 350

d) 350x = 50

= 350

13. Ravi bought 12 pencils for ₹96. Tick (✓) the equation that can be used to find the cost of one pencil. x

a) 12 + x = 96

b) 12x = 96

d) 96x = 12

= 96

14. A water tank contains 15 litres of water. Water is being added at a rate of 3 litres per minute. Tick (✓) the equation that can be used to find after how many minutes the tank will have 27 litres of water. 3

+ 15 = 27

a) 15 + 3x = 27

b) 15x + 3 = 27

d) 27x = 15 + 3

Simple Equations

15. The temperature in Shimla in the morning was 12°C. It dropped by 2.5°C every hour. Tick (✓) the equation that can be used to find after how many hours the temperature would be –3°C. Choose the equation that can be used to solve the problem: a) 12 – 2.5x = –3

c) –2.5x = 12 – 3

b) 12 + 2.5x = –3

d) 2.5 – x = –3

16. The sum of two numbers is 45. If one number is twice the other, find the numbers. Show your work here. 1) Let the smaller number be x. Then the larger number will be 2x. 2) As per the given condition: x + 2x = 45 3) 3x = 45 4) ... 5) ... Answer:

Simple Equations

divide both sides by 3

17. The sum of two numbers is 51. One number is 3 more than twice the other. Find the numbers. Show your work here.

18. Two numbers differ by 7. One half of the larger number plus 3 equals the smaller number. Find the numbers. Show your work here.

Simple Equations

19. An alloy consists of five parts nickel and four parts iron. How much of each element is present in an alloy weighing 1440 grams? Show your work here.

20. An alloy consists of two parts gold, one part platinum and ten parts silver. How much of each metal is present in an alloy weighing 910 grams? Show your work here.

Simple Equations

21. A cleaning solution consists of seven parts vinegar, five parts baking soda and four parts water. If the total mass of the solution is 1280 grams, how much of each substance is present? Show your work here.

22. Anita, Ramesh and Priya together read 72 pages of a book. Priya read 5 times as many pages as Anita and Ramesh read twice as many pages as Anita. How many pages did Priya read? Show your work here.

Simple Equations

6. ANGLE PAIRS AND PARALLEL LINES ANGLE PAIRS !

Unless stated otherwise, any angle read with a protractor must be recorded to the nearest 1°.

1. Use a protractor to measure each angle. Write the measurements in the correct boxes.

Angle Pairs and Parallel lines

2. Look at the angles below. Do as instructed. a) Measure x. b) Construct a 30° angle that shares the vertex and one ray with angle x. c) Compute x + 30° d) Construct a 45° angle from angle y sharing the same base ray. Calculate y + 45°.

x + 30° =

y + 45° =

Angle Pairs and Parallel lines

3. Look at each angle and its measurement. Use a protractor and check if its measurement is correct. If not, write the correct measurement.

93°

30° E=

146° 25° G=

4. Find the complementary angle and supplementary angle for each angle given below. Angle measure

Complementary angle

Supplementary angle

34° 67° 102° 145° 89° If an angle cannot not have its complement, write “Does not exist”.

Angle Pairs and Parallel lines

5. Look at all the angles having vertex O. For each given pair of angles, write whether they are complementary, supplementary or neither. C

a) AOB and BOC: b) AOB and BOD: c) BOC and COD: d) AOC and COD: e) List all supplementary pairs in the image.

Angle Pairs and Parallel lines

6. Solve. a) An angle A is 30° greater than its complement. Find angle A and its complement.

b) An angle B is 18° less than its supplement. Find angle B and its supplement.

c) Two complementary angles C and D are in the ratio 2:3. Find angles C and D.

d) Two supplementary angles E and F differ by 56°. Find angles E and F.

e) Angles G = 2x + 7° and H = 5x − 1° are complementary. Find x, angle G and angle H.

Angle Pairs and Parallel lines

7. Angles AOB and BOC form a linear pair. Use this information to answer the following questions. B

a) Given that AOB = 2x + 18° and BOC = 4x – 12°, find x.

b) Find: AOB =

and BOC =

c) Find the complement of the smaller of the two angles.

d) A ray OD bisects BOA. Find the measures of the two angles formed at OD:

Angle Pairs and Parallel lines

8. Angle AOB is right angled. Use this information to answer the given questions. B

a) Suppose AOC = 2y + 40° and COB = y – 4°, find y.

b) Find AOC =

and BOC

c) Are AOC and COB complementary or supplementary?

d) If the ray OC is rotated so that AOC decreases by 12°, find the new measures. of AOC =

and COB =

Angle Pairs and Parallel lines

9. Look at the given image and answer the questions that follow.

y z

ℓ

x O

a) Given that x = 3y + 3° and z = 4y + 1°, find

y, and

b) Find the supplements and complements of angles x, y and z. Complete the given table.

Angle measure

Complementary angle

Supplementary angle

x y z If an angle cannot not have its complement, write "Does not exist".

Angle Pairs and Parallel lines

10. See the figure below. Answer the questions that follow. D

a) AOC and COB from a linear pair. If AOC = 64°, find COB = b) BOD and AOD form a linear pair. If BOD = 38°, find DOA = c) Do COD and DOA form a linear pair? Tick (✓) the correct answer: Yes

d) List all distinct linear pairs in the image.

Angle Pairs and Parallel lines

. .

11. In each part, angles A and B form a linear pair. Find x and then find the measures of both angles. a) A = 4x + 12°,

B = 2x + 48°

b) A = 3x + 15°,

B = 5x + 45°

c) A = 7x + 14°,

B = 2x + 22°

d) A = 9x – 9°,

B = 3x + 15°

Angle Pairs and Parallel lines

12. Look at the given image and answer the questions that follow.

B C

ℓ

a) If A = 2x + 6°, B = 3x + 24°, C = x + 30°, find x.

a) Find: A =

, B=

b) Which angle is the largest?

Angle Pairs and Parallel lines

, C=

13. Rays OA and OB point in the opposite directions. Use this information to answer the following questions. P

a) If AOP = 3x + 6° and POB = 96°, find x .

b) Find: AOP =

and POB =

c) Which of the two is acute? Which is obtuse? Acute:

, Obtuse:

Angle Pairs and Parallel lines

14. In each figure, the two angles A and B share a common arm. Their other two sides form a pair of opposite rays. The measure of A is correct. Check whether the measure of B is correct. a)

B = 145°

A = 45°

Is the measure of B correct? Yes

A = 70°

Is the measure of B correct?

Yes

If not, write the correct value:

B = 135°

A = 45°

Is the measure of B correct? Yes

If not, write the correct value:

B = 120°

Angle Pairs and Parallel lines

B = 110°

A = 70°

Is the measure of B correct? Yes

If not, write the correct value:

15. Look at the following image. Answer the questions that follow. ℓ2

B C

A O

ℓ1 D

a) If A = 3x + 16° and C = 5x – 8°, find x.

b) Find: A=

, B=

, C=

and D =

c) List all pairs of vertical angles (vertically opposite):

d) A ray OE bisects B. Find the measures of the two angles formed by OE:

Angle Pairs and Parallel lines

16. Look at the following image. Answer the questions that follow. ℓ2

B C

a) If

A O

B = 2y + 12° and C = y + 24°, find y.

b) Find: A= Acute:

, B=

, C=

, Obtuse:

c) Which of the angles are acute? Which are obtuse?

ℓ1

Angle Pairs and Parallel lines

and D =

17. Look at the image below. A measures 68°. Answer the following questions. ℓ2

B C

a) Find: B =

, C=

A O

ℓ1

, D=

b) Find the sum of the two acute angles at O.

c) Find A + B = d) By how many degrees does the obtuse angle exceed the acute angle at this intersection?

Angle Pairs and Parallel lines

18. Look at the angles formed at point O. Answer the questions given below.

B A

C D

Let A = 2k + 20°, B = k + 16°, C = 3k + 4°, D = 2k + 10°. a) If E is 10° greater than A, find k.

b) Find: A = D=

, B=

, C=

and E =

c) Which angle is the largest? Which are acute? Largest angle: d) Find: C + E =

Angle Pairs and Parallel lines

Acute angles:

19. Look at the given angle. Do as instructed. B

a) Measure

AOB =

b) Construct a 30° angle sharing vertex O and the base ray OA. Compute AOB + 30° = c) Construct a 45° angle sharing the same base ray OA. Compute

AOB + 45° =

(d) Classify each resulting sum as acute, right, obtuse, or straight: AOB + 30° is

AOB + 45° is

Angle Pairs and Parallel lines

20. See the figure below. Answer the questions that follow. C

O a) Measure:

AOB =

, BOC =

b) Compute: AOB + BOC = c) Compute: | AOB – BOC| = (d) Compute: 180° − ( AOB + BOC) = State whether this angle in (d) is acute or obtuse. Tick (✓) the correct answer: Acute

Obtuse

21. Three angles are defined as P = 2x + 14°, Q = 3x + 16° and R = x + 20°. Answer the questions below. a) If P + Q = 120°, find x, then find P and Q.

Angle Pairs and Parallel lines

b) Compute: P + R =

c) For what value of x is Q + R = 180°?

d) For x from part (a), is P < Q? Tick (✓) the correct answer: Yes

22. In the given figure, AOD is a straight line. AOB = 3y + 12°, BOC = 2y – 6° and COD = 4y + 3°. Answer the following questions. C

Angle Pairs and Parallel lines

a) Find y.

b) Find: AOB =

, BOC =

and COD =

c) Which angle is the largest?

PARALLEL LINES 23. Two parallel lines ℓ₁ and ℓ₂ are cut by a transversal t. For each given pair of angles, write their relationship: corresponding, alternate interior, alternate exterior, co-interior or none. t D C F G

Angle Pairs and Parallel lines

E H

A B

ℓ1

ℓ2

a) A and E: b) B and F: c) C and G: d) D and H: e) B and E: f) A and G: g) C and E: h) D and F:

24. Look at the figure from the previous question. For each type of angle relationship, list all the unique pairs. a) All pairs of corresponding angles:

b) All pairs of alternate interior angles:

c) All pairs of alternate exterior angles:

d) All pairs of co-interior angles:

Angle Pairs and Parallel lines

25. In the following figure, two parallel lines ℓ₁ and ℓ₂ are cut by transversal t. Use it to answer the questions. t

D C

F G

E H

a) If A = 4x + 8° and E = 2x + 38°, find x.

b) Find: A =

and E =

c) Find: B = d) Find: F =

Angle Pairs and Parallel lines

and

A B

ℓ1

ℓ2

26. In the following figure, two parallel lines ℓ₁ and ℓ₂ are cut by transversal t. Use it to answer the questions. t D C

F G

E H

A B

ℓ1

ℓ2

a) Given that C = 3x + 10° and E = 2x + 40°, find x.

b) Find: C =

and E =

c) Find: D =

and H =

Angle Pairs and Parallel lines

27. In the following figure, two parallel lines ℓ₁ and ℓ₂ are cut by transversal t. Use it to answer the given questions. t D C

F G

E H

a) Given that B = 6x + 12° and E = 3x + 6°, find x.

b) Find: B =

and E =

c) Find: A =

and G =

Angle Pairs and Parallel lines

A B

ℓ1

ℓ2

28. Three parallel lines ℓ1 ǁ ℓ2 ǁ ℓ3 are cut by a single transversal t. Angle A measures 38°. Answer the following questions. t D C F G K L

A B

E H

ℓ1

ℓ2

ℓ3

At the top intersection, the marked angle A equals 38°. a) Find: B =

, C=

b) Find: E =

, F=

c) Find: J =

, L=

, D=

Angle Pairs and Parallel lines

100

29. In the given figure AB ǁ CD. Answer the given questions. A

a) If ABC = 3x + 24° and BCD = 6x + 12°, find x.

b) Find: ABC =

and BCD =

c) State which one is acute and which is obtuse. Acute angle:

Obtuse angle:

d) Find the alternate interior angle of ABC on line CD. What is its measure?

101

Angle Pairs and Parallel lines

30. Look at each figure. Answer the questions that follow. a) Are the lines m and n parallel? Tick (✓) the correct answer. Yes

No t A = 148°

E = 138°

b) Are the lines m and n parallel? Tick (✓) the correct answer. Yes

B = 75° F = 105°

Angle Pairs and Parallel lines

102

31. In each figure, ℓ₁ and ℓ₂ are parallel lines. The measure of B is correct. Check if the measure of the C is correct. If not, write the correct measure. t

a) Is the measure of C correct? Yes

B = 70°

The correct measure of C =

ℓ₁ C = 120°

ℓ2

a) Is the measure of C correct? Yes

The correct measure of C =

ℓ₁

B = 96° C = 86°

ℓ2

C =28°

a) Is the measure of C correct? Yes

ℓ₁

The correct measure of C = B =152°

103

Angle Pairs and Parallel lines

ℓ2

7. TRIANGLES AND THEIR PROPERTIES UNDERSTANDING TRIANGLES AND THEIR PROPERTIES 1. The measurements of angles/sides of some triangles are given below. Classify each triangle by its sides and by its angles. Tick (✓) the correct answer. a) A = 60°, B = 60°, C = 60°

e) Sides: 7, 24, 25

equilateral / isosceles / scalene

By sides:

By angles:

b) A = 30°, B = 60°, C = 90°

f) Sides: 9, 10, 12

equilateral / isosceles / scalene

acute / right / obtuse By sides:

By angles:

c) A = 70°, B = 70°, C = 40°

g) Sides: 13, 14, 15

equilateral / isosceles / scalene

acute / right / obtuse By sides:

By angles:

d) A = 100°, B = 40°, C = 40°

h) Sides: 9, 10, 10

equilateral / isosceles / scalene

acute / right / obtuse By sides:

By angles:

acute / right / obtuse

acute / right / obtuse By sides:

By angles:

acute / right / obtuse

Triangles and Their Properties

104

2. The measurements of three angles of some triangles are given below. Classify each triangle by its sides and by its angles. Tick (✓) the correct answers. a) 35°, 55°, 90°

d) 60°, 60°, 60°

equilateral / isosceles / scalene

By sides:

By angles:

b) 50°, 50°, 80°

e) 45°, 45°, 90°

equilateral / isosceles / scalene

acute / right / obtuse / impossible

By sides:

By angles:

acute / right / obtuse / impossible

By sides:

By angles:

acute / right / obtuse / impossible

c) 20°, 30°, 130° By sides:

equilateral / isosceles / scalene By angles:

acute / right / obtuse / impossible

3. Write the correct term related to a triangle for each description. Use: median, centroid, altitude, orthocentre. a) A line segment drawn from a vertex to the midpoint of the opposite side:

b) The point where the three medians meet:

105

Triangles and Their Properties

c) A perpendicular line segment drawn from a vertex to the opposite side:

d) The point where the three altitudes meet:

e) The point that divides each median in the ratio 2:1:

f) In a right triangle, this point coincides with the vertex at the right angle:

4. In triangle ABC, point D is the midpoint of BC and point E is the midpoint of AC. The medians AD and BE intersect at G. Use this information to answer the questions. C

D G

a) If AD = 21 cm, find: AG =

and GD =

b) If AG = 12 cm, find: GD =

and AD =

c) If BE = 30 cm, find: BG =

and GE =

Triangles and Their Properties

106

RIGHT-ANGLED TRIANGLES 5. For each triangle, the orthocentre H is shown. Answer the questions below. Choose the correct options and fill in the blanks. a) Where is the orthocentre H located?

(inside / at a vertex / outside)

What type of triangle is it? (acute / right / obtuse)

H B

b) Where is the orthocentre H located? (inside / at a vertex / outside)

What type of triangle is it? (acute / right / obtuse)

c) Where is the orthocentre H located? (inside / at a vertex / outside)

C A

What type of triangle is it? (acute / right / obtuse)

107

Triangles and Their Properties

6. In triangle ABC, the points D, E and F are the midpoints of BC, CA and AB. The medians of the triangle meet at G. Use this information to answer the questions below. C

D G

a) Given: AG = 10 cm and GE = 7 cm, find: AD =

and BE =

b) Given: AD = 27 cm and BG = 18 cm, find: AG =

, GD =

BE =

7. Solve. a) In

ABC with AB =AC, if A = 36°, find B and C.

b) In

ABC with AB = AC, if B = 52°, find A and C.

c) In

ABC with BC = CA, if B = 38°, find A and C.

Triangles and Their Properties

108

8. ABC is a triangle. In each of the following, find its angles. a) A = 62°, B = 47°, C = b) A = 35°, B = c) A =

. , C = 85°.

, B = 73° , C = 62°.

d) Angles are in the ratio A : B : C = 3 : 5 : 7. Find each angle.

e) Angles are in the ratio A : B : C = 2 : 2 : 1. Find each angle.

f) A = x + 14°, B = 2x – 9°. Find C.

109

Triangles and Their Properties

9. In triangle ABC, side AC is extended beyond C to form an exterior angle at C. Answer the following questions. a) A = 45° and B = 63°. Find the exterior angle at C and the interior angle C.

b) The exterior angle at C is 118° and A = 50°. Find B and C.

A = 2x°, B = (x + 28)° and the exterior angle at C is 136°. Find x and all the

angles of the triangle.

10. Which image correctly shows an exterior angle at C? Tick (✓) the correct options. C

B Image 1

B Image 2

B Image 3

Triangles and Their Properties

110

11. In ABC, side AB equals side AC. Side BA is extended beyond point A to form an exterior angle at A. Use this to answer the following questions. C

a) If the exterior angle at A equals 132°, then: B=

and C =

b) If B = С and the exterior angle at A equals 118°, find all the angles.

c) If A = (2k + 6)° and the exterior angle at A equals (5k + 6)°, find k

and all the angles.

111

Triangles and Their Properties

12. Read each statement. Tick (✓) whether it is true (T) or false (F). If false, write the correct statement. a) In a triangle, an exterior angle equals the sum of the two interior angles, one of which is adjacent to it.

True

False

Correct: b) In a triangle, if an exterior angle is 120°, then the two interior angles opposite to it must each be 120°.

True

False

Correct: c) In any triangle, each exterior angle is greater than any interior angle of the triangle.

Correct:

True

False

d) In an isosceles triangle, the exterior angle at the top vertex angle equals the sum of the two base angles.

True

False

Correct: e) In a triangle, if two exterior angles are equal, then the interior angles at those vertices are equal.

True

False

Correct: f) In a triangle, if A = B, then the exterior angles at A and B are equal. True

False

Correct:

Triangles and Their Properties

112

13. In the figure, points A, C and D are collinear. Answer the following questions. A

C B D a) ABC = (2x + 10)° and BCD = (x + 25)°. Find ACB in terms of x. ABC = b) The exterior angle at C for triangle BCD is 52°. Find BCD. BCD =

14. Look at triangle ABC in the figure. Its sides are extended at vertices B and C to form exterior angles. Answer the following questions.

113

Triangles and Their Properties

a) Given that A = (x + 20)°. The exterior angle at B is (3x − 10)°. The exterior

angles at B and C are equal. Find x and B.

b) The exterior angles at B and C are equal. The exterior angle at C is (y + 46)° and A = 64°. Find y and C.

15. In ABC, sides AC and AB are extended as shown. Answer the following questions.

C A

a) Given that A = 48° and AB = AC. Find B and C. B=

; C=

b) The exterior angle at A is (x + 70)°, C = (2x − 16)° and AB = AC. Find x and all

the angles.

Triangles and Their Properties

114

16. Answer the following questions. a) In

ABC, AB = AC and A = 48°. Find the exterior angles at B and C.

b) In triangle ABC, BC = CA. The exterior angle at A is 138°. Find all the interior angles.

17. Can a triangle with the given sides be formed? Tick (✓) the correct answer. a) 5, 7, 11:

Yes

c) 8, 8, 16:

b) 6, 9, 14:

Yes

d) 10, 12, 21:

Yes Yes

No No

18. In each of the following, two sides a and b of a triangle are given. Write all integer values of the third side c such that a triangle can be formed.

a) a = 7, b = 12, values of c:

b) a = 9, b = 15, values of c: c) a = 11, b = 18, values of c:

19. Find the hypotenuse of each right triangle. Fill in the blanks.

115

a) A right triangle has legs 6 cm and 8 cm. The hypotenuse is

cm.

b) A right triangle has legs 9 cm and 12 cm. The hypotenuse is

cm.

Triangles and Their Properties

20. Find the missing leg of each right triangle. Fill in the blanks. a) A right triangle has hypotenuse 13

cm and one leg 5 cm. The other leg is cm.

b) A right triangle has hypotenuse 25 cm and one leg 7 cm. The other leg is cm.

21. Classify each triangle by its angles. Hint: Use Pythagoras theorem. If the square of the largest side = the sum of the squares of the other two sides, the triangle is right-angled.

If the square of the largest side > the sum of the squares of the other two sides, the triangle is obtuse-angled.

If the square of the largest side < the sum of the squares of the other two sides, the triangle is acute-angled. a) Sides: 7, 24, 25. Type:

b) Sides: 5, 6, 7. Type:

Triangles and Their Properties

116

c) Sides: 8, 15, 18. Type:

22. Answer the following questions.

a) Rectangle ABCD has sides 9 cm and 12 cm long. Find the length of its diagonal.

b) Rectangle SPQR has one side 8 cm long and a diagonal 17 cm long. Find the length of its other side.

23. A ladder of length 13 m leans against a wall. Its foot is 5 m from the wall. Find the height of the top of the ladder above the ground.

13 m

117

Triangles and Their Properties

24. A triangle ABC is right angled at B. If AB = 9 cm and BC = 12 cm, find its perimeter. A

CONGRUENCE 25. Look at the two congruent triangles. Their equal parts are marked. What is the correspondence of their vertices? Which criterion of congruence do they follow? Fill in the blanks. C

Correspondence of vertices: A

, C

Congruence criterion:

Triangles and Their Properties

118

26. Look at the two congruent triangles. Their equal parts are marked. What is the correspondence of their vertices? Which criterion of congruence do they follow? Fill in the blanks. C

Correspondence of vertices: A

, C

Congruence criterion:

27. Do the given side lengths determine congruent triangles? Tick (✓) the correct answers. a)

ABC: AB = 7 cm, BC = 9 cm, AC = 12 cm. DEF: DE = 7 cm, EF = 9 cm, DF = 12 cm. Yes

ABC: AB = 6 cm, BC = 10 cm, AC = 13 cm

DEF: DE = 6 cm, EF = 10 cm, DF = 12 cm Yes

119

Triangles and Their Properties

28. Look at the two congruent triangles. Their equal parts are marked. What is the correspondence of their vertices? Which criterion of congruence do they follow? Fill in the blanks.

Correspondence of vertices: A

, C

Congruence criterion:

29. Look at the two congruent triangles. Their equal parts are marked. Which criterion of congruence do they follow? Fill in the blank.

Congruence criterion:

Triangles and Their Properties

120

30. Two congruent triangles are shown below and one pair of their equal sides is marked. Can you determine which other sides are equal? Explain your answer. C F

Answer:

31. Triangles ABC and DEF are congruent. AB corresponds to DE and BC corresponds to EF. If DE = 5 cm and DF = 21 cm, find AC. AC =

cm.

32. Look at the given figure. Find the value of x. C 16 cm

121

Triangles and Their Properties

D 7 cm

33. In triangles ABC and PQR: AB = PQ = 7 cm BC = QR = 9 cm AC = PR = 11 cm a) Are the triangles congruent? If yes, which criterion proves it? b) If B = 65°, find Q.

c) Write all pairs of congruent angles.

34. In triangle ABC, AB = AC. The point D is the midpoint of BC. The point A is joined to the point D. a) Are triangles ABD and ACD are congruent? State which criterion you used. b) If ABD = 55°, find ACD. c) What can you say about ADB and ADC?

35. Triangles PQR and XYZ are congruent with the correspondence P X, Q Y, R Z. a) If PQ = 8 cm, QR = 10 cm and XZ = 13 cm, find PR. b) If P = 50° and Q = 70°, find Y and Z.

c) If the perimeter of triangle PQR is 31 cm, find the perimeter of triangle XYZ.

36. In triangles ABC and DEF, AB = DE, A = D and AC = DF. a) If AB = (2x + 3) cm, DE = 15 cm, find x. b) If A = (3y – 10)° and D = 80°, find y.

c) If BC = 12 cm, find EF.

Triangles and Their Properties

122

8. RATIO AND PROPORTION RATIO AND PROPORTION 1. Write each statement in three different forms: as a fraction, as a ratio (with a : symbol), and in words. The first one is done for you. a) 4 apples to 6 apples:

4 6

b) 6 apples to 9 apples:

= 4 : 6 = 4 to 6.

c) 15 pencils to 10 pencils:

d) 24 students to 36 students:

2. Read each statement. Fill in the blanks. Write the answers in simplest form. a) There are 20 boys and 30 girls in a school. The ratio of boys to girls is b) There are 48 students in total. Among them, 18 study music. The ratio of students who study music to all students is

c) A garden has 12 apple trees, 24 orange trees and 36 banana trees.

123

i. The ratio of apple trees to banana trees is

ii. The ratio of banana trees to total trees is

Ratio and Proportion

3. Express each ratio in its simplest form. a) 12 : 18

c) 64 : 80

b) 35 : 49

d) 150 : 225

: :

4. Find the given ratios. Write each in its simplest form. a) 3 m to 250 cm

b) 4 kg to 750 g

c) 2.5 L to 1500 mL

d) 45 minutes to 2 hours

Ratio and Proportion

124

5. Are the following pairs of ratios equivalent? Tick (✓) the correct answer. a) 8 : 12 and 10 : 15

Yes

b) 21 : 35 and 9 : 15

Yes

c) 45 : 60 and 9 : 12

Yes

d) 16 : 24 and 20 : 30

Yes

6. Simplify each ratio to its lowest terms. a) 84 : 126

b) 144 : 180

c) 315 : 525

d) 128 : 226

7. Simplify the following ratios to their lowest terms. a) 18 : 27 : 45 b) 72 : 108 : 180 c) 150 : 225 : 300

125

Ratio and Proportion

: :

8. Find the given ratios. Write each in its simplest form. a) 2 m 50 cm to 150 cm :

b) 3 kg 600 g to 1 kg 200 g :

c) 1.5 L to 2500 mL :

d) 90 minutes to 2 hours 15 minutes :

9. A recipe uses flour, sugar and butter in the ratio 3 : 2 : 1. If 450 g of flour is used, how much sugar and butter are needed?

Ratio and Proportion

126

10. Compare the ratios. Write >, < or = in each circle. a) 7 : 9

5:6

b) 16 : 24

3:5

c) 14 : 21

20 : 30

d) 9 : 20

11 : 25

11. Basket A has red and blue beads in the ratio 24 : 18. Basket B has them in the ratio 35 : 28. Which basket has a greater ratio of red to blue beads? Tick (✓) the correct answer.

Basket A

Basket B

12. Rank the following ratios from smallest to greatest. Write 1 for the smallest, 2 for the next smallest, and so on. a) 5 : 12

rank

c) 9 : 16

rank

b) 7 : 15

rank

d) 11 : 18

rank

13. Divide each quantity into the given ratio. Write each part.

127

a) Divide 180 into 2 : 4.

b) Divide 350 into 3 : 4.

c) Divide 560 into 5 : 9.

Ratio and Proportion

14. A prize of ₹720 is shared among three friends A, B and C in the ratio 2 : 3 : 7. How much money does each friend receive?

, B:

, C:

15. The sides of a rectangle are in the ratio 3 : 5. If the perimeter of the rectangle is 64 cm, find its length and breadth.

Length:

; Breadth:

16. If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.

Answer:

Ratio and Proportion

128

17. If P : Q = 3 : 7 and Q : R = 2 : 9, find P : Q : R.

Answer:

18. Three numbers are in the ratio 2 : 3 : 5. If their sum is 200, find the numbers.

Numbers:

19. The monthly incomes of A and B are in the ratio 7:9. If A's income is ₹1,40,000, what is the income of B?

Income of B:

129

Ratio and Proportion

20. The angles of a triangle are in the ratio 2 : 3 : 4. Find the three angles.

Angles:

°,

21. Check if the following ratios form a proportion. Tick (✓) the correct option. a) 7 : 9 and 21 : 27

Yes

b) 25 : 35 and 15 : 20

Yes

c) 14 : 18 and 21 : 28

Yes

22. Find the value of x in each proportion. a) 6 : 9 = 10 : x

c) x : 36 = 5 : 12

b) 15 : x = 25 : 40 x =

23. The sides of two triangles are proportional. The first triangle has sides 9 cm, 12 cm and 15 cm. The second triangle has a side of 18 cm that corresponds to the 9 cm side. Find the remaining sides of the second triangle.

Sides:

cm,

cm.

Ratio and Proportion

130

24. On a map, 1 cm represents 5 km. If two towns are 7.2 cm apart on the map, find the actual distance between them.

Answer:

25. Find the number that must be added to both terms of the ratio 5 : 7 so that it becomes 3 : 4.

Answer:

26. Find the number that must be subtracted from both terms of the ratio 33 : 53 so that it becomes 3 : 5.

Answer:

131

Ratio and Proportion

27. Find the number which must be added to both terms of the ratios 3 : 5 and 7 : 11 so that the new ratios are equal.

Answer:

28. Find the mean proportional between 9 and 25.

Answer:

29. Two rectangles have their corresponding sides in the ratio 3 : 4. If the smaller rectangle has a side length of 12 cm, find the corresponding side length of the larger rectangle.

Answer:

30. Three numbers: a, b, c are said to be in continued proportion if a : b = b : c. If 4, 6, and z are in continued proportion, find z.

Answer:

Ratio and Proportion

132

31. The cost of 4 notebooks is ₹56. Use ratios to find the cost of 7 notebooks.

Answer: ₹

32. At a store 2.5 kg of sugar cost ₹150. Use ratios to find the cost of 4 kg of sugar.

Answer: ₹

33. A machine produces 360 parts in 6 hours at a constant rate. How many parts will it produce in 10 hours at the same rate?

Answer:

34. 12 workers can finish a job in 8 days. How many days will 6 workers take to finish the same job, working at the same rate?

Answer:

133

Ratio and Proportion

days

35. A car travels 180 km in 3 hours at a constant speed. How far will it travel in 5 hours at the same speed?

Answer:

36. A cyclist covers 84 km in 4 hours at a constant speed. Find the speed in km/h.

Answer:

km / h

37. A car travels at 54 km/h. Find its speed in m/s.

Answer:

m/s

Ratio and Proportion

134

38. An athlete covers a distance of 1500 m in 5 minutes. Find his speed in km/h.

Answer:

km / h

39. A car travels 60 km at 40 km/h and then next 90 km at 60 km/h. Find the average speed for the whole journey.

Answer:

km / h

40. A cyclist travels for 2 hours at 12 km/h and then for 3 hours at 8 km/h. Find her average speed.

Answer:

135

Ratio and Proportion

km / h

41. A bus travels the first 60 km at 30 km/h, the next 60 km at 40 km/h, and the final 75 km at 25 km/h. Find its average speed for the whole journey.

Answer:

km / h

42. Two cars are 300 km apart. They start travelling towards each other. Their speeds are 50 km/h and 70 km/h. How many hours will it take them to meet?

Answer:

hours

43. A man is walking at a speed of 6 km/h. A train 120 m long and moving in the same direction passes him in 12 seconds. Find the speed of the train in km/h.

Answer:

km / h

Ratio and Proportion

136

44. Two cyclists start together and travel in the same direction. One travels at 18 km/h and the other at 12 km/h. How long will it take for the faster cyclist to be 24 km ahead of the slower one?

Answer:

hours

45. A train leaves City A at 7:30 am and reaches City B at 10:00 am. The cities are 150 km apart. Find the average speed of the train.

Answer:

km / h

46. Two towns X and Y are 300 km apart. A bus leaves Town X at 6:00 a.m. and travels towards Town Y at 40 km/h. Another bus leaves Town Y at 7:30 a.m. and travels towards Town X at 60 km/h. At what time will the two buses meet?

Answer:

137

Ratio and Proportion

47. A car has a mileage of 20 km per litre. How many litres of fuel are needed to travel 260 km?

Answer:

48. 12 workers can finish a job in 10 days. How many days will 15 workers take to finish the same job, working at the same rate?

Answer:

days

49. A bus covers a distance in 4 hours at 60 km/h. How long will it take to cover the same distance at 80 km/h?

Answer:

hours

Ratio and Proportion

138

50. With a certain amount of money, 30 toys can be bought at ₹120 per toy. How many toys can be bought with the same amount of money if the price drops to ₹80 per toy?

Answer:

51. State whether each situation shows direct proportion or inverse proportion. Tick (✓) the correct option. Then solve the question. a) For 24 students, the canteen prepares 24 cups of juice. How many cups are needed for 30 students? Direct

Inverse

Answer: b) An athlete takes 1.5 hours to complete a track at a speed of 8 km/h. How long will he take if he runs at 10 km/h? Direct

Answer:

139

Ratio and Proportion

Inverse

52. Three litres of paint cover a wall having an area of 200 ft². How many litres of paint are required to cover a wall having an area of 350 ft²?

Answer:

53. A team of 12 workers completes 720 units of work in 10 days. If the workload increases to 900 units and the team size increases to 15 workers, how many days will it take to complete the work?

Answer:

days

Ratio and Proportion

140

9. PERCENTAGE, PROFIT AND LOSS, DATA HANDLING SIMPLE INTEREST PERCENTAGE AND ITS APPLICATIONS 1. The pie chart shows the favourite fruits of a group of students. Use it to answer the given questions.

Apple Banana Mango

25% 35%

Orange

20% 20%

a) What fraction of the students prefer mango? Answer: b) What percentage of the students prefer orange? Answer: c) How many students prefer banana, if there are 200 students in all?

Answer:

141

Percentage, Profit and Loss, Simple Interest

2. The pie chart shows the means of transport used by students. Use it to answer the given questions.

40%

On foot Bicycle Bus

25%

15%

Car

20%

a) What is the ratio of students who go by bus to those who go by bicycle? Answer: b) What percent of students travel on foot? Answer: c) If 300 students were surveyed, how many of them travel by car?

Answer:

Percentage, Profit and Loss, Simple Interest

142

3. The pie chart shows marks obtained by a student in four subjects. Use it to answer the given questions.

Math Science

30%

25%

20%

25%

English Social Science

In which subject did the student score the highest marks? Answer: b) What percentage of the total marks was obtained in English? Answer: c) If the total marks are 400, how many marks were scored in Science?

Answer:

4. Convert each fraction into a percentage. a)

143

3 4

Percentage, Profit and Loss, Simple Interest

5. Convert each percentage into a fraction in its simplest form. a) 45%

b) 125%

6. Convert each decimal into a percentage. a) 0.07

b) 1.5

7. Express each percentage as a decimal. a) 8%

b) 150%

8. Express the following numbers as percentages of one another. a) 15 and 60 15 is

% of 60;

60 is

% of 15

b) 40 and 80 40 is

% of 80;

80 is

% of 40

Percentage, Profit and Loss, Simple Interest

144

9. Solve. a) 12 is 30% of x. Then x =

b) 48 is 120% of x. Then x =

10. Convert each into a percentage. Then arrange them in ascending order. Fill in the blanks. 0.45, 2 , 38% 5

Answer:

11. In a class of 50 students, 30 are girls and 20 are boys. a) Write the ratio of girls to boys. b) What percent of the class are girls?

145

Percentage, Profit and Loss, Simple Interest

: %

12. A rack has 100 books. Of these, 45 are storybooks and the rest are textbooks. a) Write the ratio of storybooks to textbooks.

b) What percent of the books are textbooks?

13. There are 30 members in the school eco-club. The ratio of boys to girls in the club is 2:3. a) How many boys are in the club? b) What percent of the members are girls?

14. In a survey, 70% of people preferred tea, while 30% preferred coffee. a) Write the ratio of people who prefer tea to people who prefer coffee. : b) If 200 people were surveyed, how many preferred tea?

Percentage, Profit and Loss, Simple Interest

146

15. Find the following values. a) 45% of 200:

b) 12% of 150:

16. Express the first number as a percentage of the second number. a) 40 of 160:

b) 90 of 150:

17. Solve. a) 30% of a number x is 18. What is the value of x?

Answer: b) 70% of a number x is 42. What is the value of x?

Answer:

147

Percentage, Profit and Loss, Simple Interest

18. Convert each to a percentage. a) 250 m out of 1 km

Answer:

b) 45 seconds out of 3 minutes

Answer:

19. A tank contains 800 L of water. Of this, 120 L was used for watering plants. a) What percentage of the water was used?

Answer:

b) What percentage of the water is left?

Answer:

Percentage, Profit and Loss, Simple Interest

148

20. A school has 600 students. Of these, 40% joined the Sports Club, 25% joined the Music Club, and the remaining students joined the Reading Club. a) How many students joined the Sports Club? b) How many students joined the Music Club? c) What percentage of students joined the Reading Club?

21. A book cost ₹200 last year and ₹240 this year. a) By how much did the price increase? b) By what percent did the price increase?

Answer:

22. The population of a town decreased from 50,000 to 45,000. a) By how much did the population decrease? b) By what percent did the population decrease?

Answer:

149

Percentage, Profit and Loss, Simple Interest

23. A store increased the price of a shirt from ₹400 to ₹500. Later, the price was reduced back to ₹400. a) By what percent was the price first increased?

Answer:

b) By what percent was the price then decreased?

Answer:

24. Compare and find the percent difference. a) Is 72 more or less than 80? By what percent?

Answer: b) Is 96 more or less than 80? By what percent?

Answer:

Percentage, Profit and Loss, Simple Interest

150

PROFIT AND LOSS 25. The price of a mobile phone increased by 20%. However, during the festive season, the price was reduced by 10%. If the original price was ₹10,000, answer the following: a) What is the net percentage change in the price?

Answer:

b) What is the new price of the mobile phone?

Answer:

26. A store buys a toy for ₹150 and later sells it for ₹180. a) How much profit does the store make on a toy? b) What is the profit percentage?

Answer:

151

Percentage, Profit and Loss, Simple Interest

27. A store bought a book for ₹250. The store sold it for ₹200 during the clearance sale. a) How much was the loss? b) What percent loss was made?

Answer:

28. A pen was sold for ₹96 at a profit of 20%. Find the cost price of the pen.

Answer:

29. A table was sold for ₹1,320 at a 12% loss. Find its cost price.

Answer:

Percentage, Profit and Loss, Simple Interest

152

30. A machine was purchased for ₹12,000 and later sold at a loss of 15%. Find the selling price.

Answer:

31. A shirt is marked at ₹1,500. It is sold at a discount of 10%. a) What is the discount amount? b) What is the selling price of the shirt?

Answer:

32. A jacket costs ₹800 and is marked at ₹1,200. It is sold after a discount of 20% on the marked price. a) What is the selling price?

b) What is the profit percentage?

153

Percentage, Profit and Loss, Simple Interest

33. A discount of 30% was given on the marked price of an article. It was sold for ₹840. What was the marked price?

34. An article was marked at ₹2,000. It was sold after two successive discounts — first 20% and then 10%. a) What was the single equivalent discount percent?

% b) What was the final selling price?

Percentage, Profit and Loss, Simple Interest

154

35. Selling an object for ₹176 would give the same profit as the loss from selling it for ₹144. a) What is its cost price?

Answer: b) What would be the profit percentage and the loss percentage?

Answer:

% and

36. A shopkeeper bought 40 T-shirts at ₹150 each and sold them all. His total loss was equal to the cost of 8 T-shirts. a) What was the selling price of each T-shirt?

Answer: b) What was his percentage loss?

Answer:

155

Percentage, Profit and Loss, Simple Interest

SIMPLE INTEREST 37. Find the simple interest on ₹5,000 at 8% per year for 2 years.

38. A sum of ₹3,600 gives ₹972 as simple interest in 3 years. Find the rate of interest.

39. A sum of ₹2,500 earns ₹600 as simple interest at 12% per year. Find the time required.

years

Percentage, Profit and Loss, Simple Interest

156

10. RATIONAL NUMBERS

INTRODUCING RATIONAL NUMBERS 1. Write rational numbers with the given numerators and denominators. Reduce each to standard form. a) Numerator = (–3) × 4; Denominator = 6 × 3

Standard form: b) Numerator = (–5) × (–6); Denominator = 3 × 20

Standard form: c) Numerator = 10 × (–7); Denominator = (–14) × 11

Standard form: d) Numerator = 7 × 15; Denominator = 35 × (–2)

Standard form:

157

Rational Numbers

2. Which rational numbers will fill the boxes on the number line? Choose the correct numbers and write them in the boxes. –5

-4

-3

–

-2

-1

–

5 2

3. Write each rational number as an equivalent rational number with denominator 24. a)

5 6

b) −

7 8

11 12

4. Convert each rational number to its equivalent rational numbers with the given numerators or denominators. Fill in the blanks. a)

3 5

;

b) −

7 8

5. Reduce each rational number to its standard form.

−42 56

−120

−63 −84

144

−180

Rational Numbers

158

6. Rewrite each rational number so that the denominator is positive. Then express it in standard form. a)

−98

−50 −65

7. Reduce each of the following rational numbers to standard form. a) 2

3 4

b) −3

5 6

c) 3

5 6

d) −2

9 8

8. Write the additive inverse of each rational number.

a) −

0 5

9. Write the additive inverse of each rational number.

a) The additive inverse of – b) The additive inverse of

159

Rational Numbers

13 4

– 10

is is

. .

10. Write the multiplicative inverse of each rational number. a) The multiplicative inverse of − b) The multiplicative inverse of −

7 5

−11

11. Arrange the rational numbers from smallest to largest. Show your work. 7

−

,−

4 3

,−

12. Arrange the rational numbers from smallest to largest. Show your work. −

11 12

2 3

,−

5 6

,−

7 9

Final order:

13. In each case, write the missing numerator to form an equivalent rational number. a)

−3 5

;

−9

;

−4 6

−18

Rational Numbers

160

14. Are the rational numbers in each pair equivalent? Tick (✓) the correct option. a)

−9

21 28

and

Yes

−42

−216 15. Reduce to standard form. Write the answer −378 in the blank.

a 3 and are equivalent rational b 8 numbers. If b is positive and less than 41, write two a possible values of . b 16. The numbers

Answer:

161

Rational Numbers

17. Compare the rational numbers in each pair. Write <, = or > in each box. a) −

5 6

−

c) −

−

27 30

18. Read the statement given below. Is it true or false? Tick (✓) the correct option. If two negative rational numbers have the same denominator, the number with True

the greater absolute numerator is smaller.

False

m where m is an integer. If x lies exactly 24 5 1 halfway between − and − , find the value of m. 6 4

19. Let x =

20. Use the midpoint method to generate three 1 1 rational numbers lying between and . Fill in the 6 3 blanks. r1 =

Answer: r1 =

1 1 + 6 3

r2 =

, r2 =

, r3 =

1 + r1 6

r3 =

r1 +

1 3

Rational Numbers

162

5 1 and – . Then write 8 4 two rational numbers with denominator 40 that are

21. Find the midpoint of – closest to it. Answer:

22. Write three rational numbers with denominator 1 1 36 that lie between and . 9 3 Answer:

23. Find the rational number with denominator 60 7 1 that lies exactly midway between – and – . 10 2 Answer:

24. Are the rational numbers in each pair multiplicative inverses of each other? Tick (✓) the correct option. a) − b)

163

5 3

−7 8

and − and

3 5

Rational Numbers

Yes

−9

and

−9 2

Yes

OPERATIONS ON RATIONAL NUMBERS 25. Add or subtract the rational numbers in each pair. Reduce the result to standard form. a) c)

12 11 18

+ −

12 7

b) −

d) −

10 5 8

−

+ 3 4

3 5

26. Add. Write the answer in standard form. 13 7 + 20 30

Answer:

27. Simplify. Write the answer in standard form. 17 5 − − 24 18

Answer:

28. Solve for x.

5 6

+x=

Rational Numbers

164

29. Solve for x. Write your answer in standard form. x −

11 20

=−

3 5

30. Multiply and write the result in standard form.

−7 9

15 14

;

11 12

−8

31. Divide. Reduce each result to its standard form.

−10 27

;

b) −

21 −8

32. Evaluate.

165

() ( ) 3

−4 9

−10 21

Rational Numbers

11 20 9

−

3 8

( )

÷ −

27 28

( ) 2 3

−

1 6

(

5 8

13 15

−

12 25

)

−

1 3

3 4

2 3

( )

( )( )

(

( ) 2

( ) 7

3 7

1 2

2 7

)

6 1 2

−

4 5

Use this space for calculations:

11 33. Find area of a rectangle whose sides are m 12 3 and m. 5

Answer (in standard form):

34. Solve for x. 5 8

×x=−

15 32

Answer: x =

Rational Numbers

166

2 of its capacity filled. More water is 5 1 added, filling an additional of the capacity. What 8 fraction of the capacity is filled now? 35. A tank has

Answer:

11 m long is cut into two parts in the ratio 12 2 : 1. Find the length of the longer part. 36. A rope

Answer:

7 37. The temperature of a city at midnight was – °C. 4 11 By noon, it increased by °C. What was the 8 temperature at noon?

Answer:

167

Rational Numbers

3 38. A shopkeeper decreased the price of a shirt by 10 1 of the original price. He then increased it by of the 5 reduced price. What fraction of the original price is the final price?

Answer:

84 −2 39. Simplify −126 . Is it equal to 3 ? Tick (✓) the correct option. Yes

7 40. A number is multiplied by and then multiplied 6 12 by . How many times the original number is the 7 final number? Answer (in standard form):

Rational Numbers

168

11. CONSTRUCTION OF TRIANGLES DATA HANDLING CONSTRUCTING TRIANGLES 1. A line segment AB has a point P on it. Construct the perpendicular to AB at P.

2. Construct the bisector of ∠XOY. Y

3. Read each statement. Is it true or false? Tick the correct option.

169

a) The bisector of an angle divides it into two equal angles.

True

False

b) The sum of angles around a point is 360°.

True

False

c) The three angle bisectors of any triangle meet at one point.

True

False

Construction of Triangles

4. Fill in the blanks. a) A triangle can be uniquely constructed if we know at least

elements of the triangle.

b) At least one of the known elements must be a

5. Can a unique triangle be constructed using the given elements? Tick the correct option. a) Three sides

Yes

b) Two sides and the included angle

Yes

c) Three angles

Yes

d) Angle B and sides AB and AC

Yes

e) Three sides measuring 3 cm, 3 cm, and 7 cm

Yes

6. Construct a triangle with sides 5 cm, 6 cm and 7 cm. Then classify the triangle based on its sides. Write your answer in the box.

Construction of Triangles

170

7. Construct an equilateral triangle ABC with side 5 cm and find its angles.

∠A =

;

∠B =

;

∠C =

8. The sides of triangle ABC are in the ratio 2 : 3 : 4 and its perimeter is 18 cm. Find the length of each side, then construct the triangle.

AB =

;

BC =

;

AC =

9. A student constructs triangle ABC such that AB = 7 cm, AC = 4 cm and ∠A = 45°. Which construction criterion has been used? Write your answer in the blank.

171

Construction of Triangles

10. Complete the construction of triangle ABC using the given base AB. Make ∠A = 120° and mark AC so that AB = AC. Then classify the triangle based on its angles.

B triangle.

Triangle ABC is

11. Construct a triangle ABC with given side BC, ∠B = 45°, and ∠C = 60°.

45°

60°

Measure ∠A:

12. Triangle PQR is to be constructed with ∠P = 40°, ∠Q = 70° and side PQ = 6 cm. A student instead constructs ∠R = 70° in place of ∠Q. Will this lead to an incorrect construction? Tick the correct option. Yes

Construction of Triangles

172

13. Construct a triangle ABC with given side AB, ∠A = 50°, and ∠B = 80°.

50° A

80° B

Which side is the longest?

14. In right-angled triangle MLN, the legs ML and MN are in the ratio 4 : 3, with ML = 8 cm. Construct the triangle.

Measure the length of the hypotenuse:

15. Two right-angled triangles have legs 3 cm and 4 cm, and 12 cm and 9 cm respectively. Find the ratio of their areas.

Ratio of areas

173

Construction of Triangles

16. Read each set of elements. Can a unique triangle be constructed using them? Tick the correct option. a) AB = 6 cm, BC = 7 cm, CA = 5 cm

True

False

b) AB = 5 cm, BC = 6 cm, ∠B = 120°

True

False

c) ∠A = 40°, ∠B = 60°, AB = 5 cm

True

False

d) ∠A = 60°, ∠B = 70°, ∠C = 50°

True

False

17. Read the Assertion and the Reason. Tick the correct option that best explains their relationship. Assertion (A): A triangle is uniquely determined if its three angles are known. Reason (R): The sum of the angles of a triangle is 180°. a) Both A and R are true, and R explains A. b) Both A and R are true, but R does not explain A. c) A is false, but R is true. d) Both A and R are false.

18. △ABC is to be constructed with BC = 6 cm, ∠B = 40°, and ∠C = 50°. Is this information sufficient to construct the triangle uniquely? Tick the correct option. Yes

Construction of Triangles

174

12. PERIMETER AND AREA DATA HANDLING SQUARES, RECTANGLES, TRIANGLES AND PARALLELOGRAMS 1. The side of a square measures 8 cm. Find its perimeter and area.

2. The perimeter of a square is 44 cm. Find its area.

175

Perimeter and Area

3. The area of a rectangle is 180 cm². If its one side measures 12 cm, find its perimeter.

4. The figure shown below consists of two rectangles. Its total area is 64 cm². One rectangle has a length of 10 cm and a width of 4 cm. The other rectangle has a length of 6 cm. Find the perimeter of the whole figure.

Perimeter and Area

176

5. A rectangular garden measures 18 m by 12 m. A path 2 m wide runs all around inside the boundary, as shown. Find the area of the path. 18 m

12 m

177

Perimeter and Area

6. A rectangle measures 18 cm × 12 cm. From each corner, a square of side 3 cm is cut out. Find the area of the remaining figure.

7. A square has side length of 20 cm. A smaller square of side 6 cm is cut out from it, as shown in the figure. Find the area of the remaining figure.

20 cm

6 cm

Perimeter and Area

178

10 cm

4 cm

8. A rectangular sheet of paper measures 14 cm by 10 cm. A 6 cm by 4 cm part is cut out from the inside. Find the area of the remaining sheet.

6 cm 14 cm

9. A rectangular sheet of chart paper measures 24 cm by 16 cm. A square piece of side 8 cm is cut out from the middle of the top edge, so that the sheet becomes U-shaped. Find the area and the perimeter of the remaining sheet.

24 cm

179

Perimeter and Area

16 cm

8 cm

10. A staircase-shaped figure is made of three rectangles. The bottom rectangle measures 12 cm by 4 cm, the middle rectangle 8 cm by 4 cm, and the top rectangle 4 cm by 4 cm. Find the perimeter of the figure. 4 cm 4 cm

4 cm

8 cm

4 cm 12 cm

11. In parallelogram ABCD, the height to AB is 9 cm and the height to AD is 12 cm. If the perimeter of ABCD is 70 cm, find its area.

h2 B

Perimeter and Area

180

12. In parallelogram ABCD, point P lies on AB, and a line through P parallel to AD meets DC at Q, as shown. The area of parallelogram APQD is threefifths of the area of ABCD. If AB = 24 cm, find AP. D

181

Perimeter and Area

13. The area of a triangle is 132 cm², and its base is 12 cm. Find the corresponding height.

Perimeter and Area

182

14. In the following figure, the shaded (grey) region shows the forest and the unshaded region shows the lake. Compare their areas.

15. Two triangles with bases 12 cm and 8 cm have the same height of 10 cm. Find the ratio of their areas.

183

Perimeter and Area

16. A triangle with base 12 cm and corresponding height 8 cm is divided into two smaller triangles by its median. Find the area of each smaller triangle. C

Perimeter and Area

184

17. On side AB of triangle ABC, point D is marked such that AD = 4 cm and BD = 7 cm. The area of triangle ACD is 12 cm². Find the area of triangle BCD. C

A AD = 4 cm D

185

Perimeter and Area

DB = 7 cm

CIRCLES 18. A quadrant of a circle is shaded as shown. If the radius of the circle is 7 cm, find the area and perimeter of the shaded region. D

Perimeter and Area

186

WORD PROBLEMS ON PERIMETER AND AREA 19. Four quarter-circles are cut from a square sheet of side 21 cm. Two of them have radius 7 cm each, and the other two have radius 14 cm each. Find the area of the remaining sheet.

20. A man walks once around a rectangular route with sides 14 m and 28 m. Next to him, his dog runs along a route made of six semicircles, each with diameter 14 m. How much longer is the dog’s route than the man’s?

187

Perimeter and Area

21. Adhya and Aarav run along the routes shown in the diagram. The long horizontal parts of both routes are 28 m. Adhya runs along short vertical segments of 14 m, while Aarav runs along semicircular arcs of diameter 14 m. How much longer is Aarav’s route than Adhya’s? Adhya

Aarav

Perimeter and Area

188

DATA HANDLING 13. ALGEBRAIC EXPRESSIONS GENERATING RULES IN FORMULAS AND PATTERNS 1. Continue each pattern by writing the next three terms. a) 2, 5, 8, 11,

b) 9, 12, 15, 18,

, ,

c) 7, 10, 13, 16,

. ,

d) 4, 1, −2, −5,

, ,

. .

2. The nth term of a sequence is given. Find the 21st term. a) nth Term = 2n + 1

Answer: nth Term

b) nth Term = 10n − 1

Answer: nth Term

c) nth Term = 3n − 4

Answer: nth Term

3. Find the sum of the first n natural numbers using n(n + 1) 1 + 2 + 3 + ... + n = 2

189

a) n = 10

b) n = 20

c) n = 50

Answer:

Algebraic Expressions

4. Each regular polygon below has side length a. Find the perimeter of each. a) Triangle:

c) Pentagon:

b) Square:

d) Hexagon:

5. Two rectangles have sides (l, b) and (l + 2, b − 1). Find the difference in their areas and write the result in standard form.

Answer:

6. An equilateral triangle has side 3a. A square has side 2a. Which shape has the greater perimeter? Tick the correct option. Triangle

Square

7. List the terms of each expression. a) 7x − 3y + 5

Terms:

b) −4a + 2ab − b

Terms:

Algebraic Expressions

190

ALGEBRAIC EXPRESSIONS AND TERMS 8. For each expression, write the numerical coefficient of a and the algebraic coefficient of a. a) 2.5ab

Numerical coefficient:

, Algebraic coefficient:

b) −7a

Numerical coefficient: c) −

, Algebraic coefficient:

3 Numerical coefficient:

, Algebraic coefficient:

9. Identify all constant terms in each expression. a) 5x − 3y + 9

Constant term:

b) −2m + 3mn − n2

Constant term:

c) 7 − k

Constant term:

10. Complete the factor tree of 12ab. 12ab 12

ab a

191

Algebraic Expressions

11. Complete the factor trees. c) 24abc:

a) 15ab: 15ab 15

24abc ab

abc с

b) 18a2b: 18a2b 18

a2b ab a

12. Write the expression for each statement. a) Sum of x and twice y:

Expression:

b) Three-fourths added to one-third z:

Expression:

c) One-fourth of n subtracted from three n:

Expression:

d) t decreased by five:

Expression:

13. Write the algebraic expression based on each statement. a) Perimeter of rectangle (l, b): P = b) Area of a square of side a: S = c) Perimeter of an equilateral triangle of side a: P =

Algebraic Expressions

192

14. Write an expression for each situation. a) Arjun has ₹m. He gets ₹50 more. How much does he have now? Expression:

b) Priya spends ₹x on books and ₹y on a pen. How much did she spend in total? Expression:

c) A ribbon of length r cm is cut into two parts. One part is r/3 cm. What is the length of the other part? Expression:

15. A regular polygon has n sides, and each side is a units long. Write an expression for its perimeter in terms of n and a. Expression:

16. Identify the like terms in each set. Write them in the blanks. a) 3x, −2x, 5y, x, y

Like terms:

(b) ab, −3ab, a2b, b2, 4ab

Like terms:

17. Look at the terms in each pair. Are they like terms or unlike terms? Tick the correct option. a) 7p and −3p Like

193

Unlike

Algebraic Expressions

b) 2a2 and 2ab Like

Unlike

c) xy and yx Like

Unlike

ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS 18. Simplify each expression by grouping the like terms. a) 5x − 2y + x + 3y − 4

Answer:

b) ab + 2a − 3ab + b − a

Answer:

19. Classify each algebraic expression based on the number of terms. a) 7x

c) a + b + c

b) 3y − 5

d) p − q + r − s

20. Find the odd one: x + y, a + b, p + q + r.

21. Identify all terms in each expression. Then classify it as monomial, binomial, or trinomial. a) 2m − 3n + 5

Terms:

Type:

b) −a + b

Terms:

Type:

c) 7

Terms:

Type:

22. Add. a) (3x − 2y) + (x + 5y)

Answer:

b) (a + 2b + 3) + (2a − b − 1)

Answer:

Algebraic Expressions

194

23. Add using the column method. a) 5p − 2q + 1 and −3p + 4q − 6

Answer:

b) 2m + n and m − 3n + 7

Answer:

24. Simplify. a) x + y + 2x − 3y + 4

Answer:

b) ab − 2a + 3ab + b − a

Answer:

25. Add. x − 2, 3x + 5, −2x + 1

Answer:

26. Rohan has ₹x and Meera has ₹(2x + 50). Find the total amount they have. Answer:

27. A rectangle has length l and breadth b. A square has side a. Find the sum of their perimeters. Answer:

195

Algebraic Expressions

28. Subtract. a) (5x − 3y) − (2x + y)

Answer:

b) (a + 4b − 7) − (2a − b + 1)

Answer:

29. Subtract using column method. a) (4p + q − 6) − (2p − 3q + 1)

Answer:

b) (m − 2n) − (3m + n − 5)

Answer:

30. What should be subtracted from x + 3y to get 2x − y?

Answer:

31. What should be subtracted from 5a − 2b + 7 to get 2a + 3b − 1?

Answer:

Algebraic Expressions

196

32. A shop sells ice creams. On Day 1, it earned ₹(x + 40). On Day 2, it earned ₹(x − 10). How much more did it earn on Day 1 than on Day 2?

Answer:

r + 3) cm is cut 4 from it. What length of ribbon remains? 33. A ribbon is r cm long. A piece of (

Answer:

34. Simplify the brackets. Write your answer in standard form. a) [ 2x − { x − (3 − y) } ]

Result:

b) { 3a − [ 2a − (b − 1) ] }

Result:

35. Simplify.

197

a) x − (2x − 3) + 4

Answer:

b) − { 2y − [ 3y − (1 − y) ] }

Answer:

Algebraic Expressions

36. Simplify each expression, then evaluate it for the given values. a) x + 2x − 3 at x = −2

Answer:

b) 2a − b + (a − 2b) at a = 3, b = −1

Answer:

c) 3m − 2n − (m + n) at m = −4, n = 5

Answer:

37. Evaluate each expression for the given values. a) 2p − q at p = 7, q = 3

Answer:

b) a2 − b at a = −3, b = 4

Answer:

c) xy − 2x at x = −2, y = −5

Answer:

Algebraic Expressions

198

38. Use the given information to answer each question. a) A rectangle has sides l and b. A square has side a. Find the difference between their perimeters. Write your answer in standard form. Answer: b) A square has side (k + 2). Write an expression for its area without expanding the brackets. Answer:

39. Simplify. Write the result in the standard form. (x − 3y + 1) + (2x + y − 4) − (x − 2y − 5)

Answer:

40. Classify the following polynomials. a) 9t b) u − 7 c) p + q + r d) a − b + c − d

41. Look at the terms in each pair. Are they like terms or unlike terms? Tick the correct option. a) 3xy and −5yx Like

199

Unlike

Algebraic Expressions

b) 2a2 and 2a Like

Unlike

42. Simplify each expression. a) (4x − 3y + 2) + (x + y − 5)

Answer:

b) (m − 2n) − (n − 3m + 1)

Answer:

43. A right triangle has legs b and h and hypotenuse c. Find its perimeter.

Answer:

44. Evaluate each expression for the given values of the variables. a) 2x + 3 at x = 5

Answer:

b) a2 − 2a at a = −2

Answer:

Algebraic Expressions

200

14. EXPONENTS AND POWERS EXPONENTS 1. Fill in the blanks. a) 1 lakh =

units;

b) 1 crore =

units;

c) 10,00,000 =

lakhs.

2. Match the following. a) 100 hundreds

1) 10000 units

b) 1000 tens

2) 1000000 units

c) 10 lakhs

3) 10000 units

3. Write the following in usual form. a) 5 lakh =

units;

b) 3 crore =

units;

c) 2 crore 50 lakh =

units.

4. Compute. 101 =

;

102 =

103 =

;

5. Is 100,000 equal to 10⁵? Tick the correct option. Yes

201

Exponents and Powers

6. Write as repeated multiplication. a) 25 =

b) 73 =

;

c) 104 =

;

7. Identify the base and exponent in each expression. a) 57

base:

exponent :

b) (−3)4

base:

exponent :

c) 109

base:

exponent :

8. Compute. 34 =

52 =

;

26 =

;

9. Fill in the blanks. a) Square of 11 =

;

b) Square of 12 =

;

c) Cube of 7 =

10. Compare using >, < or =. a) 25

b) 33

c) 102

210

0n =

(n > 0).

11. Fill in the blanks. a1 =

;

1n =

;

Exponents and Powers

202

12. Evaluate. −32 =

(−3)2 =

;

(−2)3 × 22 =

;

13. Fill in the blanks. 22 = 23 = 24 =

32 =

33 =

43 =

34 =

;

42 =

44 =

;

52 =

53 =

54 =

14. Are the following statements true or false? Tick the correct option. a) 23 = 32

b) 42 = 24

True

False

True

c) 53 = 125 False

True

False

15. A Bacteria double every hour. There were 8 bacteria in a sample. How many bacteria will there be after 2 hours? Answer:

16. Compute. −24 =

(−2)4 =

;

(−2)5 =

;

17. Will the result be positive or negative? Tick the correct option. a) (−7)12

b) (−5)11

203

Positive

Negative

Positive

Negative

Exponents and Powers

c) (−2)20

Positive

Negative

18. Is (−3)⁵ negative? Tick the correct option. Yes

19. Compare using >, < or =. a) 74

b) (−2)6

(−2)7

20. Write each of the following as the product of its prime factors in power notation. a) 216 =

b) 768 =

c) 1500 =

21. Write each of the following as the product of its prime factors in power notation. a) 540 =

b) 490 =

Exponents and Powers

204

22. Find the HCF and LCM of each pair. a) 23 × 32 × 5 and 22 × 33

HCF:

, LCM:

b) 24 × 72 and 25 × 3 × 7

HCF:

, LCM:

23. A school wants to divide 216 students into groups of 9. Express 216 as a product of its prime factors, then use it to find the number of groups. Prime factorization of 216 = Number of groups:

24. Write all factors of 12 as pairs of divisors (e.g. (1, 12)). Answer:

25. Write all factors of 36 as pairs of divisors. Answer:

26. Simplify. 23 × 25 =

205

Exponents and Powers

54 × 52 =

27. Simplify. (3 × 5)2 =

(2 × 7)3 =

;

28. Compute. 40 =

1000 =

;

(−3)0 =

;

29. Simplify. 27 23

;

30. Simplify:

() 3 4

;

31. Simplify. (23)4 =

(52)3 =

;

32. Simplify. 64 × 23 32 × 6

Exponents and Powers

206

33. Solve.

() 2

10 = 1000 ; Answer: x =

;t=

125

34. If 6p × 63 = 362, find p.

Answer: p =

35. Find n if x2 × xn = x7. Answer: n =

36. Write as repeated multiplication. a)

() 2

( )

b) −

37. Compute.

() 2 3

−

3 5

38. Find reciprocals of the following expressions. a)

207

() p

Exponents and Powers

b) (−a)n

() 2 5

39. Write in standard form. a) 45000 =

;

b) 8000000 =

b) 7.2 × 107 =

40. Write in usual form. a) 3.6 × 105 =

;

41. Compare using >, < or =. a) 3.8 × 107

7.2 × 106

b) 5.01 × 108

4.9 × 109

42. Find the number of digits in 8.3 × 106. Answer:

43. Multiply and write in standard form (k × 10n). (2.5 × 103) × (4 × 102) =

Exponents and Powers

208

44. The population of a country is 1.25 × 109. Express it in usual form.

45. A lab sample initially contains 9 bacteria. They triple every 2 hours. How many bacteria will there be after 6 hours? Answer:

46. A player begins with 5 points. After winning each round, the player's score doubles. Write an expression for the score after winning n rounds. Answer:

47. Given that 1 KB = 210 bytes, 1 MB = 210 KB, 1 GB = 210 MB and 1 TB = 210 GB, express 1 GB and 1 TB in bytes using powers of 2. Answer: 1 GB =

209

Exponents and Powers

bytes, 1 TB =

bytes.

15. SYMMETRY UNDERSTANDING SYMMETRY 1. Circle the figures that have only 1 line of symmetry. a)

2. Circle the figures that have exactly 2 lines of symmetry. a)

3. Circle the figures that have exactly 3 lines of symmetry. a)

4. Circle the figures that have exactly 4 lines of symmetry. a)

Symmetry

210

5. Draw the lines of symmetry for each figure. a)

6. For each figure, tick the line that shows its axis (line) of symmetry. a) 3 2

1 b)

211

Symmetry

7. In each of the following, the two figures are mirror images of each other. Identify the line of symmetry (mirror line) between them. a)

1 2

8. In which drawing are the two figures mirror reflections of each other about a line passing between them? a)

9. Reflect the quadrilateral across the red line and across the green line. Draw both reflected images.

B A

C D

Symmetry

212

10. Draw the symmetrical half of each image about the blue line.

213

Symmetry

11. Each shape can be rotated from one red line to the next. Find the angle of rotation between consecutive red lines. a)

90°

12. Which of the following shapes have line symmetry? Draw all possible lines of symmetry for each shape that has them. a)

Symmetry

214

16. VISUALISING SOLID SHAPES 3-D SHAPES AND 2-D SHAPES 1. Name the shapes. a)

circle d)

2. Find and separate 3-D and 2-D shapes. Complete the table. a)

2D a

215

Visualising Solid Shapes

3. Cross out the cones and circle the pyramids.

4. Name the shapes that these objects look like. 1)

circle 1)

Visualising Solid Shapes

216

5. Match the number of faces, edges and vertices to the correct shape. a)

faces: 6

edges: 12

vertices: 8

faces: 5

edges: 8

vertices: 5

faces: 8

edges: 18

vertices: 12

faces: 4

edges: 6

vertices: 4

faces: 7

edges: 15

vertices: 10

217

Visualising Solid Shapes

6. For each 3D shape, shade the correct net.

Visualising Solid Shapes

218

1. INTEGERS Page 1 1. Ascending order: −12, −8, −3, 0, 5, 7. Sum of smallest and largest integers: −5. 2. T, T, T, T. 3. a) −3 < 2; b) |−7| > |−4|; c) −10 > −15; d) 0 < |−5|; 4. –5

–3

0 1 2

4 5

Page 2 5. A

6. 7.

B C D E

–5 –2 2 3 4 a) 3; b) 3; c) 4; d) 7; e) 5. a) 1; b) −2, −1, 0, 1, 2; c) −2.

Page 3 8. a) 2; b) 9; c) 2; d) 2; e) 3. 9. a) x = –2, 2 b) x = 0. Page 4 10. a) |−4 − 5| = 9; (b) |−7 − (−1)| = 6. 11. a) −5; b) −4; c) −11; (d) 0. 12. 3 13. a) 6; b) −6; c) −12. 14. a) 23; b) 25; c) 0. Page 5 15. a) 8; b) −18. 16. a) −5; b) −5; c) −5. 17. 6°C. 18. 530 m below sea level. Page 6 19. ₹300. 20. 5. 21. a) −96; b) 42; c) −60; d) 0.

219

Answers

Page 7 22. a) −7; b) 8; c) 0. 23. 10 kg lighter. 24. Not possible. 25. b), d), e). Page 8 26. a) −14; b) −8; c) 4; d) −5. 27. a) 45; b) −8; c) 0. 28. −44 m (below start). Page 9 29. 30 floors. 30. ₹960 profit. 31. a) Commutative property of addition. 32. b) Distributive property. Page 10 33. a) Associative property, b) Commutative property, c) Distributive property. 34. [(−2) + (−5) + (−1)] + 8 = −8 + 8 =0 Properties used: Commutative Property of Addition Associative Property of Addition 35. −12.

Page 11 36. a) −7; b) 11; c) 0; d) −25. 37. a) −5; b) 9; c) 0. 38. a) 1 ; b) – 1 ; c) 1; d) – 1 4 3 10 39. a) T; b) F; c) T.

Page 12 40. a) −22; b) −30; c) −64. 41. (−8 + 6) × (−2) = 4. 42. −7. 43. 36. 44. The mistake is treating −(−5) as −5. Correct answer is –3. Page 13 45. 5. 46.

–4

–2

–3

–1

47. −4, −3, −2.

2. FRACTIONS AND DECIMALS

Page 18

Page 14 1. a) 2 = 1 ; b) 3 ; c) 5 ; 6 3 8 12

10. a) 30 ; b) 40 ; c) 45 ; d) 42 ; e) 45 ; 60 60 60 60 60 44 f) 60

2. Answers vary (student shading; check the target fraction of the area). Page 15 3. a) 4 = 2 ; b) 7 ; c) 15 = 3 . 6 3 10 20 4 4. a) 5 1 ; b) 5 1 ; c) 3 2 ; d) 4 1 ; e) 2 2 ; 5 2 3 4 8 7 f) 7 ; 12 5. a) 19 ; b) 43 ; c) 43 ; d) 8 ; e) 7 ; f) 47 6 8 3 4 10 5

11. a) 9 and 10 ; b) 8 and 7 ; 12 12 12 12 c) 25 and 12 ; d) 28 and 33 ; 40 40 36 36 12. a) 6 , 8 , 9 ; b) 24 , 25 , 28 12 12 12 40 40 40 c) 72 , 70 , 54 126 126 126

Page 16

Page 19 13. a) <; b) >; c) >; d) <; e) <.

6. a) a = 25; b) a = 3, b = 13; c) m = 10

14. a) >; b) >; c) >; d) >; e) >.

7. a) 12 = 2 ; b) 45 = 3 ; c) 16 = 2 ; 3 60 4 24 3 18

15. a) 3 < 1 < 2 < 5 8 2 3 6 b) 5 < 4 < 3 < 7 9 7 5 10

d) 28 = 2 ; e) 40 = 4 ; f) 55 ; g) 63 = 3 ; 3 50 5 81 84 4 42

h) 120 = 4 ; i) 81 = 3 5 108 4 150

Page 20 16. a)

Page 17 8. b) 10; c) 8; d) 12; e) 12; f) 15 9.

1 2

1 5

5 ; b) 1 ; c) 9 ; d) 2 ; 7 9 11 5

17. a) 1

7 ; b) 1 ; c) 37 ; d) 25 ; 12 12 40 36

18. 3 10

Page 21

19. a) 7 ; b) 8 ; c) 5 d) 29 9 8 42 6

Answers

220

Page 29 36. a) 7; b) 25 c) 6; d) 0.45; e) 6.75; f) 1; g) 6.3; h) 7.85

Page 22 20. a) 21. a)

2 ; b) 19 ; c) 3 ; 3 20 16

12 ; b) 21 ; c) 1 ; d) 3 ; 35 44 6 10

Page 23 3 3 13 1 ; b) 3 ; c) 1 ; d) 9 4 5 14 15

22. a) 1 Page 24

1 23. cup 2 24. a) 1

1 ; b) 1 29 ; c) 1 23 ; d) 128 ; 20 48 27 135

25. 6 pieces

Page 30 37. a) 1.5; b) 2.1; c) 2.2; d) 1.65; e) 6.3; f) 1.47; 38. a) 5.5; b) 3.2; c) 4.9; d) 3.75; Page 31 39. a) 8; b) 5; c) 16.5; d) 8; e) 150; f) 198; g) 700 Page 32 40. 1.5m 41. 20 apples 42. 57 km/h 43. ₹48.30

Page 25 26. a) 3

5 29 13 1 ; b) ; c) 5 ; d) 3 ; 6 32 24 9

Page 26 27. 140 pages

28. 81

29. ₹40

30. a) 0.405 < 0.45 < 0.5; b) 0.8 < 0.805 < 0.82; c) 0.12 < 0.125 < 0.13. Page 27 1 1 3 3 1 1 ; b) ; c) ; d) ; e) ; f) ; 4 8 4 5 5 2 32. a) 0.75 b) 0.7; c) 0.45; d) 0.44; e) 0.4; f) 0.125. 31. a)

33. a) 5.9; b) 4.6; c) 10; (d) 3.65. 34. a) 8.15; b) 10.6; c) 5.69; d) 12 Page 28 35. a) 2.85; b) 7.65; c) 7.9; d) 2.3; e) 10.4; f) 11.5; g) 8; h) 11.10

221

Answers

3. DATA HANDLING Page 33 1. a) 600; b) 70; c) Zebra; d) Elephant; e) 200 Page 34 2. a) Pacific; b) 11,022; c) 5,527, 7,725, 8,442, 11,022; d) Pacific; e) Atlantic Page 35 3. (b) 4. 30 5. 6 6. 10.25 Page 36 7. 122 8. 35 9. 77 10. 8 11. 21

Page 37 12. a) 48; b) 2.4; c) 1.6 13. 68 14. 41 Page 38 15. a) 528; b) 132; c) 73; d) 18.25; e) 612 f) 153 Page 39 16. c d a b 17. 3 18. 2, 5, 8 19. 12, 15, 20, 20, 25, 30, 35, 35, 35; mode: 35 Page 40 20. 12, 15, 20, 20, 25, 30, 30, 35, 35; modes: 20, 30, 35 35 21. a) 2; b) 7; c) 2, 3 Page 41 22. 4 23. 12, 15, 20, 20, 25, 30, 35, 35, 35; median: 25 24. 3.5 25. 12, 15, 20, 20, 25, 30, 35, 35; median: 22.5 Page 42 26. 17 27. a) 3; b) 15; c) 165 years; d) 11 years; e) 13 years; f) 12 years g) 7 years

Page 45 30. a) 71 m, b) Big Ben; c) Tower of Belem; d) Big Ben Page 46 31. a) 60 mm; b) February; c) June; d) 132mm Page 47 32.

Speed (in km/h)

350 300 350 200 150 100 50 0

falcon

gazelle

ostrich

mosquito

beetle

bee

cheetah Animals

Page 48 33.

Length (in cm)

2.5 2 1.5 1 0.5 0

ant

Insects

Thu

Days

Page 49 34.

Total number of gadgets sold

140 120

Page 43 28. a) 25 years; b) 17 years; c) cat; d) 6 years; e) rabbit

100

Page 44 29. 45 students, 6, Grade B

80 60 40

Mon

Tue

Wed

Answers

222

4. INTRODUCTION TO PROBABILITY Page 50 1. a) Impossible; b) Sure; c) Equally likely; d) Unlikely; e) Likely 2. a) Unlikely; b) Impossible; c) Equally likely; d) Likely; e) Sure

5. SIMPLE EQUATIONS Page 57 1.

Page 51 3. a) Unlikely; b) Likely; c) Impossible; d) Sure; e) Equally likely

Page 52 2 5 1 7 8 = ; c) = ; d) 4. a) 8; b) 5 20 4 20 20 3 7

6. a)

2x – 2 = 10

x = 10

3x + 4 = 16

x = 20

5x − 2 = 13

x=4

x+2=7

x=3

4x – 3 = 2x + 7

x=5

x+2=

1 2

x+5

2 (x – 3) = x + 5

x = 11 x = –30

Page 58 2.

Page 55 1 1 47 1 20 = ; b) ; c) ; d) 9. a) 5 6 100 2 100

Twice a number decreased by 4 is 10.

x + 7 = 15

Seven more than a number is 15.

3x + 8 = 20

Half of a number is equal to 6.

2x − 4 = 10

Three times a number increased by 8 equals 20.

Page 56 5 1 19 16 8 = ; b) = ; c) ; 20 4 50 50 25 2 8 = d) 5 20

x=4

Page 54

10. a)

x–7=3

1 4 1 5 3 = ; b) = ; c) 4 12 3 12 12

3 6 = 7. 5 10 11 1 8. = 22 2

x=6

Page 53 5.

x+5=9

One third of a number increased by 5 is 11. Three quarters of a number minus 2 equals 7. Five sixths of a number plus 4 is equal to 9. Seven tenths of a number decreased by 3 equals 8.

5 6 3 4 7

10 1 3

x=6

x+4=9 x–2=7 x–3=8 x + 5 = 11

Three-fourths of a number plus 12 equals 500.

2x –

Seven-tenths of a number added to 150 equals 850.

Answers

Twice a number minus one-fifth of that number is equal to 200.

Half of a number decreased by 25 is equal to 150.

223

10 3 4

x – 25 = 150 1 5

x = 200

x + 150 = 850 x + 12 = 500

Page 59 5 3 8 3. a) x = 3; b) x = ; c) x = ; d) x = 3 4 5 Page 60 4. a) x = 6; b) x = 5; c) x = –4; d) x = 3

Page 74 21. Vinegar: 560 g, Baking soda: 400 g, Water: 320 g 22. 45

Page 61 4 3 6 3 ; b) x = ; c) x = ; d) x = 5. a) x = 3 2 5 2 Page 62 3 3 2 6. a) x = ; b) x = –2; c) x = – ; d) x = – 2 4 5

6. ANGLE PAIRS AND PARALLEL LINES

Page 63 8 9 2 7. a) x = ; b) x = ; c) x = ; d) x = 3 3 2 3

Page 75 1. A = 28°, B = 68°, C = 118°, D = 156°.

Page 64

Page 76

26 13 2 8. a) x = 1; b) x = ; c) x = ; d) x = 3 4 5 Page 66 9. a) x = 4; b) x = 7; c) x =–5; d) x = 6 Page 67 15 5 1 10. a) x = ; b) x = ; c) x = ; d) x = 19 4 2 2 Page 68 1 36 8 11. a) x = 8; b) x = ; c) x = ; d) x = 5 5 9 Page 70 12. b) 13. b)

14. a)

Page 71 15. a) 16. 15 and 30 Page 72 17. 16 and 35

18. 20 and 13

Page 73 19. Nickel: 800 g, Iron: 640 g 20. Gold: 140 g, Platinum: 70 g, Silver: 700g

2. x = 62°, x + 30° = 92°, y = 43°,

y + 45° = 88°

Page 77 3. E = 35°, F = 90°, G = 140°, H = 25°. 4. a) 34°: comp. = 56°, sup. = 146°. b) 68°: comp. = 22°, sup. = 112°. c) 102°: comp. = no, sup. = 78°. d) 145°: comp. = no, sup. = 35°. e) 89°: comp. = 1°, sup. = 91°. Page 78 5. a) AOB and BOC: complementary.

b) AOB and BOD: supplementary. c) BOC and COD: neither. d) AOC and COD: supplementary. e) ( AOB, BOD); ( AOC, COD)

Page 79 6. a) A = 60°, comp. = 30°. b) B = 81°, supp. = 99°. c) C = 36°, D = 54°. d) E = 118°, F = 62°. e) G = 31°, H = 59°.

Answers

224

Page 80 7. a) x = 29°. b) AOB = 76°, BOC = 104°. c) comp. = 14°. d) BOD = AOD = 38°.

Page 88 15. a) x = 12. b) A = 52°, B = 128°, C = 52°, D = 128°. c) Vertical pairs: ( A, C) and ( B, D). d) 64° and 64°.

Page 81 8. a) y = 18°. b) AOC = 76°, BOC = 14°. c) Complementary. d) AOC = 64°, BOC = 26°.

Page 89 16. a) y = 48.

Page 82 9. a) x = 69°, y = 22°, and z = 89°. b) x: comp. = 21°, supp. = 111°. y: comp. = 68°, supp. = 158°. z: comp. = 1°, supp. = 91°. Page 83 10. a) COB = 116°; b) DOA = 142°; c) No; d) AOC and COB; BOD and DOA. Page 84 11. a) x = 20; A = 92°, B = 88°. b) x = 15; A = 60°, B = 120°. c) x = 16; A = 126°, B = 56°. d) x = 14.5; A = 121.5°, B = 58.5°.

225

b) A = 72°, B = 108°, C = 72°, D = 108°. c) Acute: A, C; obtuse: B, D. Page 90 17. a) B = 112°, C = 68°, D = 112°. b) 136°; c) 180°; d) 44°. Page 91

18. a) k = 28. b) A = 76°, B = 44°, C = 88°, D = 66°, E = 86°. c) Largest: C. Acute: all five angles. d) 174°. Page 92 19. a) AOB = 58°.

b) 88°; c) 103°; d) 88° — acute; 103° — obtuse

Page 85 12. a) x = 20. b) A = 46°, B = 86°, C = 50°. c) B.

Page 93 20. a) AOB = 38°, BOC = 110°; b) 148°. c) 72°; d) 32°; Acute. 21. a) x = 18; P = 50°, Q = 70°. b) 88°; c) x = 36; d) Yes.

Page 86 13. a) x = 26. b) AOP = 84°, POB = 96°. c) Acute: AOP; obtuse: POB.

Page 94 22. a) y = 19. b) AOB = 69°, BOC = 32°, COD = 79°. c) COD.

Page 87 14. a) No; correct B = 135°. b) No; correct B = 110°. c) Correct; d) Correct;

Page 95 23. a) Corresponding b) Alternate interior c) Corresponding

Answers

d) Alternate exterior. e)same-side interior f)alternate exterior g)alternate interior h)corresponding

Page 96 24. a) Corresponding: A and E, B and H, C and G, D and F. b) Alternate interior: B and F, C and E. c) Alternate exterior: A and G, D and H. d) Same-side interior: B and E, C an F. Page 97 25. a) x = 15°. b) A = 68°, E = 68°. c) B = 112°. d) F = 112°, H = 112°. Page 98 26. a) x = 30°. b) C = 100°, E = 100°. c) D = 80°, H = 80°. Page 99 27. a) x = 18°. b) B = 120°, E = 60°. c) A = 60°, G = 60°. Page 100 28. a) B = 142°, C = 38°, D = 142°. b) E = 38°, F = 142°. c) J = 38° and L = 38°. Page 101 29. a) x = 16°. b) ABC = 72°, BCD = 108°. c) Acute: ABC; Obtuse: BCD. d) 72°. Page 102 30. a) No; b) Yes. Page 103 31. a) No, C = 110°. b) No, C = 96°. c) Yes, C = 28°.

7. TRIANGLES AND THEIR PROPERTIES Page 104 1. a) equilateral; acute b) scalene; right. c) isosceles; acute.

d) isosceles; obtuse. e) scalene; right. f) scalene; acute. g) scalene; acute. h) isosceles; acute.

Page 105 2. a) scalene; right. b) isosceles; acute. c) scalene; obtuse.

d) equilateral; acute. e) isosceles; right.

3. a) median. b) centroid. c) altitude.

d) orthocenter. e) centroid. f) orthocenter.

Page 106 4. a) AG = 14 cm, GD = 7 cm. b) GD = 6 cm, AD = 18 cm. c) BG = 20 cm, GE = 10 cm. Page 107 5. a) inside; acute. b) at a vertex; right. c) outside; obtuse. Page 108 6. a) AD = 15 cm, BE = 21 cm. b) AG = 18 cm, GD = 9 cm, BE = 27 cm. 7. (a) B = C = 72°. (b) A = 76°, C = 52°. (c) A = B = 38°, C = 104°.

Page 109 8. a) C = 71°. b) B = 60°. c) A = 45°. d) 36°, 60°, 84°. e) 72°, 72°, 36°. f) C = 175° – 3x.

Answers

226

Page 110 9. a) exterior at C = 108°; C = 72°. b) B = 68°, C = 62°. c) 3x + 28 = 136 Þ x = 36; then A = 72°, B = 64°, C = 44°.

Page 115 18. a) c = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

10. Image 1 and Image 3

c) c = 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28

Page 111 11. a) B = C = 66°. b) A = 62°, B = C = 59°. c) k = 24; A = 54°, B = C = 63°. Page 111 12. a) F. Correct: In a triangle, an exterior angle equals the sum of the two opposite interior angles.

b) F. Correct: If an exterior angle is 120°, then the sum of the two non-adjacent interior angles is 120°. c) F. Correct: In any triangle, each exterior angle is greater than each of the two non-adjacent interior angles (but not necessarily greater than the adjacent interior angle) d) T. e) T. f) T. Page 113 13. a) ACB = 155° – x. (b) BCD = 128°. 14. a) x = 44; B = 58°. b) y = 76; C = 58°.

Page 114 15. a) C = 66°, B = 66°. b) x = 34; A = 76°, B = C = 52°. Page 115 16. a) exterior at B = 114°; exterior at C = 114°. b) A = 42°, B = 42°, C = 96°. 17. a) Yes; b) Yes; c) No; d) Yes.

227

Answers

b) c = 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23

19. a) 10 cm; b) 15 cm. Page 116 20. a) 12 cm; b) 24 cm. 21. a) right; b) acute; c) obtuse. Page 117 22. a) 15 cm; b) 15 cm 23. 12 m. Page 118 24. 36 cm.

F, C

D; Criterion: SAS.

D, B E, C 26. A 27. a) Yes; b) No.

F; Criterion: ASA.

25. A

E, B

Page 119

28. A

D, B

E, C

F; Criterion: AAS

Page 120 29. SSS. Page 121 30. No. Knowing only that AB = DE does not tell us the correspondence of the other vertices. We cannot determine which other sides are equal without more information. 31. AC = 7 cm. 32. DB = 9 cm. Page 122 33. a) Yes, by SSS criterion b) Q = 65° c) A = P, B = Q, C = R

Page 122 34. a) Yes. AB = AC (given), BD = DC (D is midpoint), AD = AD (common). By SSS criterion, ABD ACD b) ACD = 55° (CPCT) c) ADB = ADC = 90°

Page 127 10. a) < b) > c) = d) > 11. Basket A 12. 5 : 12 1, 7 : 15 2,

35. a) PR = XZ = 13 cm b) Y = Q = 70°, Z = R = 180° – 50° – 70° = 60° c) Perimeter of XYZ = 31 cm

Page 128 14. A: ₹120, B: ₹180, 15. 20 cm, 12 cm 16. 8 : 12 : 15 Page 129 17. 6 : 14 : 63

36. a) 2x + 3 = 15, 2x = 12, x = 6 b) 3y – 10 = 80, 3y = 90, y = 30 c) EF = BC = 12 cm (CPCT)

Page 123

2. (a) 2 : 3

(b) 3 : 8

(c-i) 1 : 3

b) 150, 200

c) 200, 360

C: ₹420

18. 40, 60, 100

19. ₹1,80,000

Page 130 20. 40°, 60°, 80° 21. a) Yes b) No c) No 22. a) 15 b) 24 c) 15 23. 24 cm, 30 cm

8. RATIO AND PROPORTION 6 1. b) = 6 : 9 = “6 to 9” 9 15 c) = 15 : 10 = “15 to 10” 10 24 d) = 24 : 36 = “24 to 36” 36

11 : 18 4 13. a) 60, 120

9 : 16

Page 131 24. 36 km

25. 1 26. 3

Page 132 27. 1 28. 15

(c-ii) 1 : 2

Page 124 3. a) 2 : 3 b) 5 : 7 c) 4 : 5 d) 2 : 3 4. a) 6 : 5 b) 16 : 3 c) 5 : 3 d) 3 : 8 Page 125 5. a) Yes b) Yes c) Yes d) Yes 6. a) 2 : 3 b) 4 : 5 c) 3 : 5 d) 64 : 113 7. a) 2 : 3 : 5 b) 2 : 3 : 5 c) 2 : 3 : 4 Page 126 8. a) 5 : 3 b) 3 : 1 c) 3 : 5 d) 2 : 3 9. Sugar: 300 g, Butter: 150 g

29. 16 cm

Page 133 31. ₹98 32. ₹240

30. 9

33. 600

34. 16 days

Page 134 35. 300 km

36. 21 km/h

37. 15 m/s

Page 135 38. 18 km/h

39. 50 km/h

40. 9.6 km/h

Page 136 41. 30 km/h

42. 2.5 hours

43. 42 km/h

Page 137 44. 4 hours

45. 60 km/h

46. 9:54 am

Answers

228

Page 138 47. 13 L 48. 8 days

49. 3 hours

Page 139 50. 45 toys 51. a) Direct; 30 cups b) Inverse; 1.2 hours Page 140 52. 5.25 L

53. 10 days

9. PERCENTAGE, PROFIT AND LOSS, SIMPLE INTEREST Page 141 1. a)

1 ; b) 20%; (c) 70. 5

Page 142 2. a) 4 : 5; (b) 40%; (c) 45. Page 143 3. a) Science; (b) 20%; (c) 120. 4. a) 75%; (b) 28%. Page 144 5. a)

9 5 ; b) . 20 4

6. a) 7%; (b) 150% 7. a) 0.08; (b) 1.5. 8. a) 25%; 400%; b) 50%; 200%. Page 145 9. a) 40; b) 40 10. 38% < 40% < 45% Page 146 11. a) 3 : 2; b) 60% 12. a) 9 : 11; b) 55% 13. a) 12; b) 60%. 14. a) 7 : 3; b) 140.

229

Answers

Page 147 15. a) 90; b) 18. 16. a) 25%; b) 60%. 17. a) 60; b) 60. Page 148 18. a) 25%; (b) 25%. 19. a) 15%; (b) 85%. Page 149 20. a) 240; b) 150; (c) 35%. 21. a) ₹40; b) 20%. 22. a) 5000; b) 10%. Page 150 23. a) 25%; b) 20%. 24. a) 72 is 10% less than 80; b) 96 is 20% more than 80. Page 151 25. a) 8% increase; b) ₹10,800. 26. a) ₹30; b) 20%. Page 152 27. a) ₹50; (b) 20%. 28. ₹80 29. ₹1,500. Page 153 30. ₹10,200 31. a) ₹150; b) ₹1,350. 32. a) ₹960; b) 20%. Page 154 33. ₹1,200 34. a) 28%; b) ₹1,440 Page 155 35. a) ₹160; b) 10% and 10%. 36. a) ₹120; (b) 20%. Page 156 37. ₹800 38. 9% per year 39. 2 years

10. RATIONAL NUMBERS

Page 162

Page 157 1. a) − 2 ; b) 1 ; c) 5 ; d) − 3 . 3 2 11 2

17. − 5 < − 3 ; 7 < 5 ; − 9 = − 27 . 4 12 8 10 30 6

Page 158 2. –5 –4 –3 –2 –1 0 1 2 3 4 5 − 9 2

1 2

− 5 2

7 2

3. a) 20 ; b) − 21 ; c) 22 . 24 24 24 21 21 12 , ; b) − , − 35 . 4. a) 24 40 20 35 5. a) − 3 ; b) − 3 ; c) 3 ; d) − 4 . 4 4 5 4 Page 159 6. a) − 2 ; b) 10 . 7 13 11 ; b) − 23 ; c) 23 ; d) − 25 . 7. a) 6 6 8 4 13 0 11 ; b) − ; c) . 8. a) 4 1 6 9. a) 13 ; b) 9 . 10 14 Page 160 10. a) − 5 ; b) 11 . 7 12 5 4 <− <− 9 <− 7 . 11. − 4 8 8 3 7 3 2 11 <− <− < < 5 . 12. − 9 8 3 6 12 13. a) −15;

b) −21;

c) 12.

Page 161 14. a) Yes;

b) Yes;

c) Yes.

15. 4 . 7 16. 6 , 9 , 12 , 15 . 16 24 32 40

18. Yes. 19. m = −13 20. r1 = 1 , 4

(so x = − 13 ). 24 r2 = 5 , r3 = 7 . 24 24

Page 163 21. − 18 , − 17 . 40 40 22. 5 , 6 , 7 . 36 36 36 23. − 36 . 60 24. a) Yes;

b) No;

c) Yes.

Page 164 25. a) 11 ; b) − 3 ; c) 19 ; d) − 11 . 10 54 8 12 53 . 26. 60 27. − 71 . 72 28. x = − 1 . 4 Page 165 29. x = − 1 . 20 30. a) − 5 ; b) − 2 . 9 6 1 3 . 31. a) − ; b) 6 4 32. a) 31 ; b) 7 ; c) − 2 ; d) − 2 ; 40 7 3 36 e) 5 ; f) 11 ; g) 2 ; h) 25 ; 12 5 42 36 i) 17 ; j) 7 ; k) 2 ; l) 3 . 10 3 11 30

Answers

230

Page 174 16. a) Yes; b) Yes; c) Yes; d) No 17. c) 18. Yes

Page 166 33. 11 m2. 20 34. – 3 . 4 Page 167 35. 21 . 40

36. 11 m. 18

37. − 3 C. 8

39. Yes.

40. 2.

Page 168 38. 21 . 25

12. PERIMETER AND AREA Page 175 1. Perimeter = 32 cm; Area = 64 cm2. 2. 121 cm2. Page 176 3. Other side = 15 cm; Perimeter = 54 cm. 4. Perimeter = 40 cm. Page 177 5. 104 m².

11. CONSTRUCTION OF TRIANGLES Page 169 3. a) True; b) True; c) True. Page 170 4. a) 3; b) side. 5. a) Yes; b) Yes; c) No; d) No; e) No. 6. Scalene. Page 171 7. ∠A = 60°, ∠B = 60°, ∠C = 60°. 8. AB = 4 cm, BC = 6 cm, AC = 8 cm. 9. SAS. Page 172 10. Obtuse

11. ∠A = 75°. 12. Yes Page 173 13. AC (opposite the largest angle B = 80°).

14. 10. 15. 1 : 9.

231

Answers

Page 178 6. Area 180 cm2. 7. 364 cm². Page 179 8. Area 116 cm2. 9. Area 320 cm2; Perimeter 96 cm. Page 180 10. Perimeter = 48 cm. 11. 180 cm2. Page 181 12. Area(APQD) = AP = 3 5 Area(ABCD) AB 3 × 24 = 14.4 cm. 5 Page 182 13. Height = 2 × 132 = 22 cm. 12

AP =

Page 183 14. Their areas are equal. 15. 3 : 2. Page 184 16. 24 cm2. Page 185 17. 21 cm2. Page 186 18. Shaded area = 38.5 cm² Perimeter of shaded region = 25 cm.

Page 191 10.

ab b

a Page 192 11. a)

15ab 15

Page 187 19. 56 cm2. 20. Man: 84 m; dog: 132 m; difference: 48 m. Page 188 21. Adhya: 112 m; Aarav: 128 m; difference: 16 m.

12ab

ab a

18a2b a2b

18 a

13. ALGEBRAIC EXPRESSIONS Page 189 1. a) 14, 17, 20; b) 21, 24, 27; c) 19, 22, 25; d) −8, −11, −14. 2. a) 43; b) 209; c) 59. 3. a) 55; b) 210; c) 1275. Page 190 4. a) 3a; b) 4a; c) 5a; d) 6a. 5. 2b − l − 2. 6. Triangle 7. a) 7x, −3y, 5; b) −4a, 2ab, −b; c) 3p2q, −2q, 9. Page 191 8. a) Numerical: 2.5; Algebraic: b. b) Numerical: −7; Algebraic: 1. c) Numerical: − 1 . ; Algebraic: b. 3 9. a) 9; b) none; c) 7.

24abc 24

abc ab a

с b

12. a) x + 2y; b) z + 3 ; c) 3n − n ; 3 4 4 d) t − 5. 13. a) 2l + 2b; b) a2; c) 3a. Page 193 14. a) m + 50; b) x + y; c) 2r . 3 15. na. 16. a) x-terms: 3x, −2x, x; y-terms: 5y, y. b) ab-terms: ab, −3ab, 4ab. 17. a) Like; b) Unlike; c) Like.

Answers

232

Page 194 18. a) 6x + y − 4; b) −2ab + a + b. 19. a) monomial; b) binomial; c) trinomial; d) quadrinomial. 20. Odd: p + q + r; Reason: trinomial. 21. a) Terms: 2m, −3n, 5; Type: trinomial. b) Terms: −a, b; Type: binomial. c) Terms: 7; Type: monomial. 22. a) 4x + 3y; b) 3a + b + 2. Page 195

23. a) 2p + 2q − 5; b) 3m − 2n + 7. 24. a) 3x − 2y + 4; b) 4ab − 3a + b. 25. 2x + 4. 26. 3x + 50. 27. 2l + 2b + 4a. Page 196 28. a) 3x − 4y; b) −a + 5b − 8. 29. a) 2p + 4q − 7; b) −2m − 3n + 5. 30. −x + 4y. 31. 3a − 5b + 8. Page 197 32. 50. 33. 3r − 3. 4

34. a) x − y + 3; b) a + b − 1. 35. a) −x + 7; b) 2y − 1. Page 198 36. a) −9; b) 12; c) −23. 37. a) 11; b) 5; c) 14. Page 199 38. a) 2l + 2b − 4a; b) (k + 2)2. 39. 2x + 2. 40. a) monomial; b) binomial; c) trinomial; d) quadrinomial. 41. a) Like; b) Unlike. Page 200

233

Answers

42. a) 5x − 2y − 3; b) 4m − 3n − 1. 43. b + h + c. 44. a) 13; b) 8; c) 1.

14. EXPONENTS AND POWERS Page 201 1. a) 100,000 or 105; b) 10,000,000 or 107; c) 10. 2. 100 hundreds 10000 1000 tens 1000000 10 lakhs 10000 3. a) 5,00,000; b) 3,00,00,000; c) 2,50,00,000. 4. 101 = 10; 102 = 100; 103 = 1000. 5. Yes. Page 202 6. a) 2 × 2 × 2 × 2 × 2; b) 7 × 7 × 7; c) 10 × 10 × 10 × 10. 7. a) base 5, power 7; b) base −3, power 4; c) base 10, power 9. 8. 81; 25; 64. 9. a) 121; b) 144; c) 343. 10. a) >; b) >; c) <. 11. a; 1; 0. Page 203 12. −9; 9; −32. 13. 4, 8, 16; 9, 27, 81; 16, 64, 256; 25, 125, 625. 14. a) False; b) True; c) True. 15. 8 × 2 × 2 = 32. 16. −16; 16; −32. 17. a) Positive; b) Negative; c) Positive.

Page 204 18. Yes. 19. a) >; b) >. 20. a) 23 × 33; b) 28 × 3; c) 22 × 3 × 53. 21. a) 22 × 33 × 5; b) 2 × 5 × 72. Page 205 22. a) HCF = 22 × 32 = 36, LCM = 23 × 33 × 5 = 1080; b) HCF = 24 × 7 = 112, LCM = 25 × 3 × 72 = 4704. 23. 216 = 23 × 33; groups = 24. 24. Pairs for 12: (1, 12), (2, 6), (3, 4). 25. Pairs for 36: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6). 26. 28 = 256; 56 = 15625. Page 206 27. 152 = 225; 143 = 2744. 28. 1; 1; 1. 29. 24 = 16; 54 = 625.

Page 208 39. a) 4.5 × 104; b) 8 × 106. 40. a) 360000; b) 72000000. 41. a) >; b) <. 42. 8, 300, 000 7 digits. 43. 1.0 × 106. Page 209 44. 1, 250, 000, 000. 45. 243. 46. Score after n rounds = 5 × 2n. 47. 1 GB = 230 bytes; 1 TB = 240 bytes.

15. SYMMETRY Page 210 1. a; c. 2. a. Page 211 5.

3. b.

4. a; c; d.

31. 212 = 4096; 56 = 15625. 32. 192.

Page 207 33. x = 3; t = 3. 34. p = 1 . 35. n = 5.

6. a) 1; b) 2; c) 3.

30. 9 ; 16. 16

()()()() b) − 3 × − 3 × − 3 . ( 5) ( 5) ( 5)

36. a) 2 × 2 × 2 × 2 ; 3 3 3 3

37. 16 ; − 27 . 81 125

n 38. a) q ; b) 1 ; c) 125 . p 8 (−a)n

()

infinite number

Page 212 7. a) 2; b) 2. 8. a, c. 9.

B A

C D

Answers

234

Page 213 10.

16. VISUALISING SOLID SHAPES Page 215 1.

circle

ellipse

oval

square

rectangle

trapezium

a, f, g, h

b, c, d, e

Page 216 3. Page 214 11. 90° a)

12.

45°

36° c)

4. A. 1.Triangle 2. Circle 3.Square 4.Rectangle B. 1. Cube 2. Pyramid 3. Sphere 4. Cone

Page 217 5. a) 4; b) 3; c) 1; d) 4; e) 2.

235

Answers

22,5°

18°

90°

Page 218 6. a) 1; b) 2; c) 2; d) 1; e) 2.