Class 6 Mathematics - BeTOPPERS IIT Foundation Series - 2022 Edition

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IIT FOUNDATION Class VI

MATHEMATICS

© USN Edutech Private Limited The moral rights of the author’s have been asserted. This Workbook is for personal and non-commercial use only and must not be sold, lent, hired or given to anyone else.

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:

USN Eductech Private Limited Hyderabad, India.

PREFACE Our sincere endeavour in preparing this Book is to enable students effectively grasp & understand the Concepts of Mathematics and help them build a strong foundation in this Subject. From among hundreds of questions being made available in this Book, the Student would be able to extensively practice in each concept exclusively, throughout that Chapter. At the end of each Chapter, two or three Worksheets are provided with questions which shall cover the entire Chapter, helping each Student consolidate his / her learning. This Book help students prepare for their respective Examinations including but not limited to i.e. CBSE, ICSE, various State Boards and Competitive Examinations like IIT, NEET, NTSE, Science Olympiads etc. It is compiled by our in-house team of experts who have a collective experience of more than 40 years in their respective subject matter / academic backgrounds. This books help students understand concepts and their retention through constant practice. It enables them solve question which are ‘fundamental / foundational’ as well questions which needs ‘higher order thinking’. Students gain the ability to concentrate, to be self-reliant, and hopefully become confident in the subject matter as they traverse through this Book. The important features of this books are: 1.

Lucidly presented Concepts: For ease of understanding, the ‘Concepts’ are briefly presented in simple, easy and comprehensible language.

2.

Learning Outcomes: Each chapter starts with ‘Learning Outcomes’ grid conveying what the student is going to learn / gain from this chapter.

3.

Bold-faced Key Terms: The key words, concepts, definitions, formulae, statements, etc., are presented in ‘bold face’, indicating their importance.

4.

Tables and Charts: Numerous strategically placed tables & charts, list out etc. summarizes the important information, making it readily accessible for effective study.

5.

Box Items: Are ‘highlighted special topics’ that helps students explore / investigate the subject matter thoroughly.

6.

Photographs, Illustrations: A wide array of visually appealing and informative photographs are used to help the students understand various phenomena and inculcate interest, enhance learning in the subject matter.

7.

Flow Diagrams: To help students understand the steps in problem-solving, flow diagrams have been included as needed for various important concepts. These diagrams allow the students visualize the workflow to solve such problems.

8.

Summary Charts: At the end of few important concepts or the chapter, a summary / blueprint is presented which includes a complete overview of that concept / chapter. It shall help students review the learning in a snapshot.

9.

Formative Worksheets: After every concept / few concepts, a ‘Formative Worksheet’ / ‘Classroom Worksheet’ with appropriate questions are provided from such concept/s. The solutions for these problems shall ideally be discussed by the Teacher in the classroom.

10. Conceptive Worksheets: These questions are in addition the above questions and are from that respective concept/s. They are advised to be solved beyond classroom as a ‘Homework’. This rigor, shall help students consolidate their learning as they are exposed to new type of questions related to those concept/s.

11. Summative Worksheets: At the end of each chapter, this worksheet is presented and shall contain questions based on all the concepts of that chapter. Unlike Formative Worksheet and Conceptive Worksheet questions, the questions in this worksheet encourage the students to apply their learnings acquired from that entire chapter and solve the problems analytically. 12. HOTS Worksheets: Most of the times, Summative Worksheet is followed by an HOTS (Higher Order Thinking Skills) worksheet containing advanced type of questions. The concepts can be from the same chapter or as many chapters from the Book. By solving these problems, the students are prepared to face challenging questions that appear in actual competitive entrance examinations. However, strengthening the foundation of students in academics is the main objective of this worksheet. 13. IIT JEE Worksheets: Finally, every chapters end with a IIT JEE worksheet. This worksheet contains the questions which have appeared in various competitive examinations like IIT JEE, AIEEE, EAMCET, KCET, TCET, BHU, AIIMS, CBSE, ICSE, State Boards, CET etc. related to this chapter. This gives real-time experience to students and helps them face various competitive examinations. 14. Different Types of Questions: These type of questions do appear in various competitive examinations. They include:

• Objective Type with Single Answer Correct

• Non-Objective Type

• Objective Type with > one Answer Correct

• True or False Type

• Statement Type - I (Two Statements)

• Statement Type - II (Two Statements)

• MatchingType - I (Two Columns)

• MatchingType - II (Three Columns)

• Assertion and Reasoning Type

• Statement and Explanation Type

• Roadmap Type

• FigurativeType

• Comprehension Type

• And many more...

We would like to thank all members of different departments at BeTOPPERS who played a key role in bringing out this student-friendly Book. We sincerely hope that this Book will prove useful to the students who wish to build a strong Foundation in Mathematics and aim to achieve success in various boards / competitive examinations. Further, we believe that as there is always scope for improvement, we value constructive criticism of the subject matter, as well as suggestions for improving this Book. All suggestions hopefully, shall be duly incorporated in the next edition. Wish you all the best!!!

Team BeTOPPERS

CONTENTS 1.

Number System – I

..........

01 - 24

2.

Number System – II

..........

25 - 60

3.

Ratio and Proportion

..........

61 - 70

4.

Algebra

..........

71 - 80

5.

Geometry

..........

81 - 116

6.

Mensuration

..........

117 - 132

7.

Data Handling

..........

133 - 152

8.

Key and Answers

..........

153 - 238

CONTENTS

Chapter -1

Number System- I

Learning Outcomes

By the end of this chapter, you will understand • • • • • • • •

Comparison and Ordering of Numbers Formation of Numbers from the Digits Read, Write and Identify Large Numbers Expansion of Numbers and Place Value Indian and International System of Numeration Mathematical Operations on Large Numbers Conversion of Units Rounding of Numbers

• • • • • • •

1. Introduction Natural Numbers Natural numbers begin with 1 and continued to count. The set of numbers {1, 2, 3, 4, ...} is often referred to as the set of counting numbers. More formally, it is called the set of natural numbers. The first natural number is 1, sometimes called the unit number.

Zero There is no natural number to represent nothing, hence the need for representing nothing was realised, and that gives rise to (number) zero. The number zero denoted by the symbol ‘0’.

Estimation Strategies Use of Brackets Roman Numerals Conversion between Hindu Arabic – Roman Numerals Binary System Whole Numbers and their properties Integers and their properties

Step2: If the digits at the extreme left place in the numbers are equal in value, then compare second digits from left in the numbers. Step3:If the second digits from the left are equal, then compare the third digit from the left. Continue this process until we come across unequal digits at the corresponding places. Finally, the number with greater digit is greater the other numbers. Using Place Value Chart, numbers can be compared easily by arranging them in different places of the chart. Example: Compare the following numbers 256785623, 35621568, 25653562, 25654812, 36521568.

Whole Numbers The set of natural numbers was expanded to include zero, forming the set {0, 1, 2, 3, 4, ...}. This set is called the set of whole numbers to indicate that it is different from the set of natural numbers, which does not include zero: {1, 2, 3, 4, ...} = natural numbers {0, 1, 2, 3, 4, ...} = whole numbers

2. Comparison and Ordering of Numbers To compare given numbers, following the below given rules: Rule-1: In any two given numbers, the number which has more digits is greater than the other. Rule- 2: Suppose two or more given numbers have same number of digits, then the following steps are to be followed to compare the numbers: Step1: First compare the digits at the extreme left place in the given numbers. The number which is having greatest digit in extreme left place is greater than the other number.

T.C C T.L 2 5 6 3 5 2 5 2 5 3 6

L 7 6 6 6 5

T.TH TH 8 5 2 1 5 3 5 4 2 1

H 6 5 5 8 5

T 2 6 6 1 6

O 3 8 2 2 8

Out of the given numbers only one number has 9-digits and the remaining four are 8-digit numbers. So the number 256785623 is greater number than others. In 8-digit numbers 35621568, 36521568 are greatest as they have 3 at the crores place while each of the remaining two 8-digit numbers has 2 at the crores place. Compar e 35621568, 36521568. Clearly 36521568 > 35621568 since 36521568 has 6 at the ten lacs place and 35621568 has 5 at the same place. Finally compare 25653562, 25654812. Clearly 25654812 > 25653562 since 25654812 has 4 at the thousands place while 25653562 has 3 at the thousands place. Therefore 256785623 > 36521568 > 35621568 > 25654812 > 25653562.

6th Class Mathematics

2

Ordering of Numbers There are two ways of arrangement of numbers 1.

Ascending Order: The  arrangement  of numbers from the smallest to the greatest is known as ascending order. Ascending order means in increasing order. For example, if we have to write the numbers 81, 18, 26, and 47 in ascending order, then they will be written as 18, 26, 47, and 81.

2.

Descending Order: The  arrangement  of numbers from the greatest to the smallest is known as descending order. Descending order means in decreasing order. For example, if we have to write the numbers 54, 12, 98 and 4 in descending order, then they will be written as 98, 54, 12, 4. Ascending and descending orders are reverse of each other.

3. Formation of Numbers from the Digits The following is the Procedure to write the Smallest or Greatest Number using given Digits without Repetition:

When none of the given Digits are Zero For smallest ‘n’ digit number, arrange the given digits in ascending order, then place the ascending order of digits from extreme left most place to units place. Example: For the smallest three digit number by using the digits 5,3,9. First arrange 5,3,9 in ascending order as 3,5,9. The required smallest number is 359. For greatest ‘n’ digit number, arrange the given digits in descending order, then place the descending order of digits from extreme left most place to units place. Example: For greatest three digit number by using the digits 5,3,9. First we arrange 5,3,9 in descending order as 9,5,3. The required greatest number is 953

When one of the given Digit is Zero For smallest ‘n’ digit number, arrange the given digits in ascending order, then put zero at second place from the left and then fill the remaining places from left most place to units place by the remaining digits in ascending order. Example: The smallest three digit number formed by using the digits 0,6,2 is 206. The smallest four digit number formed by using the digits 0,5,2,8 is 2058. www.betoppers.com

For greatest ‘n’ digit number, when zero is in given digits. First put zero in the units place, then fill the remaining places from left most place to right by the remaining digits in descending order. Example: The greatest four digit number formed by using 0,4,6,9 is 9640.

Finding smallest (or Greatest) number using given digits (when repetition of digits is allowed) In this case, form the smallest (or greatest) number using the given digits without repetition. Then in the number so formed replace the smallest (or greatest) digit whose repetition is allowed. If the smallest number is to be formed with one of the digits as zero, then the number zero has to be in the second place from left. If the digit at the left most place is to be repeated twice, then the repeated digits lie on either side of zero which is in the second place when the required number is smallest number. Else zero comes after the twice repeated digits. Example: The smallest 4-digit number formed by using the digits 0,2,6 and repeating 2 twice is 2026. The smallest 4-digit number using three distinct digits when repetition of digits is allowed. We know that the smallest three distinct digits are 0, 1, 2 and they are in ascending order and if repetition of digits is allowed; the number is 1002. Here only zero can repeated twice because if 1 or 2 is repeated, then the numbers so formed are 1012, 1022. These two numbers are greater numbers than1002 and also there must be three distinct digits in required number. The greatest 3-digit number formed by using the digits 3, 9 repeating 9 twice is 993. Note: i) A number, one more than a given number is called its successor. ii) A number, one less than a given number is called its predecessor. iii) Numbers in ascending order means, the numbers from smaller to greater. iv) Numbers in descending order means, the numbers from greater to smaller. Some Noteworthy Points 1. The smallest 1-digit number is 1 and greatest 1-digit number is 9 2. The smallest 2-digit number is 10 and greatest 2-digit number is 99 3. The smallest 3-digit number is 100 and greatest 3-digit number is 999

Number System – I 4. The smallest 4-digit number is 1000 and greater 4-digit number is 9999 5. Smallest n-digit number is the number in which 1 is followed by (n - 1) zeroes. 6. Greatest n-digit number is the number in which all digits are replaced with 9’s.

4. Read, Write and Identify large numbers The number of students in your class would be a two-digit number or, at the maximum, a threedigit number. These are smaller numbers. But if you are asked the number of students in your school, then it would be a bigger number. The total number of students in a city would be a large number, i.e. at least a five-digit number. And, if we count the total number of students in the whole country, then we would have to use very large numbers (like eight or nine-digit numbers). Therefore, here, we will learn about large numbers. We are already aware of the numbers up to four digits. Now, if we add 1 to the greatest 4-digit number (i.e., 9999), then we will obtain the smallest five-digit number. This number is ten thousand (10,000). i.e. 9,999 + 1 = 10,000 (ten thousand) Similarly, 99,999 + 1 = 1,00,000 (One lac) 9,99,999 + 1 = 10,00,000 (Ten lac) 99,99,999 + 1 = 1,00,00,000 (One crore) 9,99,99,999 + 1 = 10,00,00,000 (Ten crore) and so on. We observe that, Greatest 4-digit number + 1 = smallest 5-digit number Greatest 5-digit number + 1 = smallest 6-digit number Greatest 6-digit number + 1 = smallest 7-digit number and so on. Try to read the number 7,86,790. Is it difficult? This number will be read as seven lac eighty six thousand seven hundred and ninety. Similarly, we can write the numeral value of a given number. We must remember the following conversions which will be helpful in reading and writing numbers. 1 hundred = 10 tens 1 thousand = 10 hundreds = 100 tens 1 lac = 100 thousands = 1000 hundreds 1 crore = 100 lacs = 10,000 thousands

3

Formative Worksheet 1.

Arrange the following numbers in descending order (A) 92790568 (B) 927905480 (C) 92791023 (D) 93690562 2. Which number is greater: 1 268 or 1 286? 3. Find the greater number: (A) 4387, 903 (B) 42183, 7319 (C) 80375, 154562 (D) 5483, 6109 (E) 51037, 48876 (F) 293179, 380113 4. Find the smallest number: (A) 7379, 7501 (B) 37910, 37198 (C) 417393, 417501 (D) 747631, 747609 5. Arrange the following numbers in ascending order: (A) 6895, 23787, 413752 (B) 11705, 217156, 93172, 287191 6. The smallest 4-digit number having 3 different digits ? 7. What will be the greatest and the smallest fivedigit number that can be formed using the digits 5, 7, 0, 2, and 9 without repetition? 8. Write the greatest and the smallest four-digit number. 9. Form the greatest and the smallest four-digit number using the digits 9, 3, 8, and 1 without repetition, such that 1 is at the third place of the number. 10. Arrange the following numbers in descending order: 39124, 376319, 92187, 374534 11. Write the following numbers in words and answer the questions given below. 432079, 5601729 and 1794805 (1) Which is the smallest number? (2) Which is the largest number? (3) Arrange these numbers in ascending and descending order. 12. Write four different numbers using the digits 1, 2, 5, 0, each with seven digits.

Conceptive Worksheet 1. 2. 3.

Write the greatest 6-digit number. Also write its successor. Write the smallest 7-digit number. Also write its predecessor. Write the greatest number: (A) 3255, 53215, 97501 (B) 98135, 732156, 59321 (C) 56100, 100000, 88253 (D) 36345, 63354, 36354 www.betoppers.com

6th Class Mathematics

4 4.

Write the smallest number: (A) 9212, 3921, 4456, 87123 (B) 18341, 110001, 2333, 75284 (C) 19024, 19240, 90421, 10249 5. Arrange the following numbers in ascending order: (A) 4536, 4425, 4370, 4928 (B) 25068, 25245, 25270, 25510 (C) 68952, 69825, 52896, 28956 6. Arrange the following numbers in descending order: (A) 7508, 78501, 10378, 8570 (B) 23450, 45032, 30254, 52430 (C) 110021, 201010, 22101, 102021 7. Use the given digits (without repetition) and make the greatest and the smallest 4-digit numbers: (A) 1, 3, 7, 5 (B) 2, 7, 9, 3 (C) 5, 6, 1, 0 (D) 5, 4, 0, 7 8. Write the greatest and the smallest 5-digit numbers, using the digits (digits may repeat): (A) 2 and 6 (B) 1, 3 and 7 (C) 2, 6, 4 and 8 (D) 3, 0, 5 and 7 9. Write the smallest 7-digit number, using any five different digits. 10. Write the greatest 6-digit number, using any four different digits.

5. Place Value Chart and Expansion of Numbers

104

103

102

10

25

Twenty five

625 1225 five

Six hundred twenty five One thousand two hundred twenty

65432 Sixty five thousand four hundred thirty two 875120 Eight lacs seventy five thousand one hundred twenty 9521436 Ninety five lacs twenty one thousand four hundred thirty six 98563214 Nine crores eighty five lacs sixty three thousand two hundred fourteen 869252686

Eighty six crores ninety two lacs

fifty two thousand six hundred eighty six The expanded form of few numbers is given below. 25

=

2 10 + 5 1

625

=

(6 100) + (2 10) + (5 1)

1225

=

(1 1000) +(2 100) + (2 10) + (5 1)

65432

=

(6 10000) +( 5 1000) + (4 100) + (3 10) + (2 1)

875120

=

(8 100000) + (7 10000) + (5 1000) + (1 100) + (2 10) +(0 1)

9521436 =

(9 1000000) (5 100000) + (2 10000) + (1 1000) + (4 100) + (3 10) +(6 1)

98563214 =

(9 10000000) (8 1000000) + (5 100000) + (6 10000) + (3 1000) + (2 100)+ (1 10) + (4 1)

869252686 =

(8 100000000) + (6 10000000) + (9 1000000) + (2 100000) + (5 10000) + (2 1000) + (6 100)+ (8 10) + (6 1)

1

Face Value of a Digit: The face value of a digit in a number is the value of the digit itself where ever it is placed. For example, the face value of 6 in 36528 is 6 only. Note: Place value of a digit = (face value of a digit) (value of the place) Examples: The numbers (I) 5 (ii) 25 (iii) 625 (iv) 1225 (v) 65432 (vi) 875120 (vii) 9521436 (viii) 98563214 (ix)869252686. These numbers are read as follows:

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Five

Units 1

Tens 10

105

Hundreds 100

Lacs 10000

Ten Lacs 1000000

10 6

Thousands 1000

107

Ten Thousands 10000

108

Crores 100000000

Ten Crores 100000000

Place Value System can be used to represent a number. For a given number, place values from the extreme right as units, tens, hundreds, thousands, ten thousands, lacs ten lacs, crores, ten crores etc., To write numbers, Indian place value chart can be used as below.

5

Note: Crores, lacs, thousands, one’s (hundreds, tens, ones combined is called as one’s) etc., are known as Periods. We insert comma after each Period in a number. Example: 9521436 can be written as 95,21,436

Number System – I

5 Now, we will discuss two types of number systems according to which the numbers can be written in an easier way. The two types of number systems are Indian Number System International Number System Let us now study about these. Indian Number System

6. Indian and International System of Numeration A teacher asked his students to write the number 800000000 in words.

1. 2.

Two of his students, Kunal and Arpit, wrote the number in different ways but the teacher said both are correct.

According to the Indian number system, the place value chart is divided into four groups hundred, thousand, lac, and crore. Each group is divided into subgroups as shown in the following table.

Kunal wrote the given number as eighty crore and Arpit wrote as eight hundred million. Do you know why both are correct? Kunal wrote the number according to the Indian Number System and Arpit wrote the number according to the International Number System. Periods Place holders

Crore

Lakh

Thousand

Ten crore

Crore

Ten lakh

Lakh

Ten Thousand

(T-C) 10,00,00,000

(C) 1,00,00,000

(T-L) 10,00,000

L 1,00,000

(T-Th) 10,000

Hundred

Units

Thousand

Hundred

Ten

One

(Th) 1,000

(H) 100

(T) 10

(O) 1

International Number System According to the International Number System, the place value chart is divided into groups and subgroups as follows. Periods Place holders

Hundred millions (H-m) 100,000,000

Millions Ten millions (T-m)

Millions (M)

10,000,000

1,000,000

Hundred Thousands (H-th) 100,000

Differences between the Numerals according to the Indian and the International Number System Indian Number System In this system, the first comma comes after the hundredth digit (i.e. after first three digits from the right), the second comma comes after the ten thousandth digit (i.e. after five digits from the right), the third comma comes after the ten lac digit (i.e. after seven digits from the right), and so on. For example, the number 234567890 can be written according to the Indian number system as 23,45,67,890 and can be read as twenty three crore forty five lac sixty seven thousand eight hundred and ninety.

International Number System In this system, the first comma comes after the hundredth digit (after three digits from right), the second comma comes after the hundred thousandth digit (after six digits from the right),

Thousands Ten Thousands (T-th) 10,000

Thousands (Th)

Hundreds (H)

10,000

100

Units Tens (T) 10

Ones (O) 1

the third comma comes after the thousand million digit (after nine digits), and so on. For example, according to the international number system, the number 234567890 can be written as 234,567,890 and read as two hundred thirty four million five hundred sixty seven thousand eight hundred and ninety.

Formative Worksheet 13. The difference of the place values of two 4’s in 4324 ? 14. The difference between the face values of two 4’s in 24348 is ? 15. In the number 38 417, identify (A) the place value of 8, (B) the value of the digit 3. 16. How many 3 digit numbers are there in base10 system ? 17. Write 39784012 in words using both the Indian and the International number system. 18. Insert commas between the numbers according to both the Indian and the International number system. (A) 88500784 (B) 32098175 www.betoppers.com

6th Class Mathematics

6

Conceptive Worksheet 11. What is the place value of 3 in each of the following numbers? (A) 375 (B) 2735 (C) 72953 (D) 13707 (E) 370179 12. Write the expanded form of each of the following numbers: (A) 2103 (B) 75132 (C) 107912 (D) 317956 (E)3754031 13. Insert commas to separate the periods in the Indian System of Numeration. Write and read each number: (A) 26900271 (B) 47529311 (C) 94381052 (D) 708356149 (E) 623279845 (F) 5530088 14. Insert commas to separate the periods in the International System of Numeration. Write and read each number. (A) 1974522 (B) 30706210 (C) 42358761 (D) 86465099 (E) 71531680 (F) 2795814 15. Write the following numbers in words: (A) 6,00,87,971 (B) 36,20,00,804 (C) 40,55,01,000 (D) 2,95,25,666 (E) 7,25,35,469 (F) 83,17,29,614 16. Write the following numbers in words: (A) 7,071,071 (B) 36,754,981 (C) 23,064,594 (D) 90,090,090 (E) 254,789,610 (F) 685,321,479 17. Write each of the following numbers in figures: (A) Twenty crore, twenty-nine lac, seven hundred seventeen. (B) Eight crore, thirteen lac, forty thousand, twelve (C) Ninetee crore, ninety lac, fourteen thousand, six hundred eighty.

7. Mathematical Operations on Large Numbers We have learnt addition, subtraction, multiplication and division in numbers upto 10000. We shall now learn to operate these four operations on large numbers. Let’s see few examples. Example-1: Add 39,11,207 and 7,92,316 We have 39,11,207 + 7,92,316 47,03,523 Example-2: Subtract 2,92,301 from 19,00,720 We have

19,00,720  2,92,301 16,08,419

Example-3: Multiply 91,407 by 3,147 We have

91,407 3,147 6,39,849 36,56,280 91,40,700 + 27,42,21,000 28,76,57,829

Example-4: Divide 2,63,175 by 275 We have

957 275) 263175 2475 15675 1375 1925 1925 0

(D) Fifty crore, forty lac, sixty thousand, nine. (E) Seven crore, twenty three lac, eighty-six thousand, five hundred ninety four. (F) Six million, three hundred fifty-two thousand, nine hundred forty-six. (G) Forty-nine million, seven hundred eighty-two thousand, fifty-eight. (H) Ninety million, nine (I) Twenty million, three hundred eighty thousand, one hundred. (J) Eight-one million, four hundred twelve thousand, six hundred fifty.

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8. Conversion of Units Sonia goes to a shop and asks for 10,000 grams of potatoes. The shopkeeper tells her to tell the quantity she wants in kilograms because it is very difficult for him to weigh that many grams. For this, we will first learn the various units of measurement and their appropriate usages. If we have to measure the length of a pen, then we use centimetre as the unit of measurement. However, if we have to measure the length of a pole, then centimetre will be a smaller unit of measurement. Hence, we will use metre for measuring the length of the pole.

Number System – I Similarly, we will use kilometre for measuring larger distances such as the distance between two cities. In the same manner, kilogram, gram, and milligram are used for weighing items of different weights, whereas kilolitre, litre, and millilitre are used for measuring the capacities of containers of different sizes. Let us see some conversions of unit. 1 centimetre = 10 millimetres 1 metre = 100 centimetres 1 kilometre = 1000 metres We can convert the units of weight and capacity in the same manner. For weight 1 kilogram = 1000 grams 1 gram = 100 centigrams 1 centigram = 10 milligrams For capacity 1 kilolitre = 1000 litres 1 litre = 100 centilitres 1 centilitre = 10 millilitres Now, we know that 1 kilogram = 1000 grams, i.e. 1000 grams is equivalent to 1 kilogram. Thus, Sonia should have asked for 10 kg of potatoes instead of 10,000 grams of potatoes.

Formative Worksheet 19. Find the difference between 18230 and 9108. 20. Find the product of 66 and 25 and subtract this from the smallest 5-digit number. 21. Calculate 218  57. 22. Calculate 400  16. 23. Divide 24 balloons among 6 children. 24. Divide 32486 by 214 and verify your answer using the division algorithm. 25. In a school of 3 520 students, 1 928 are boys. How many students in the school are girls? 26. In a school of 3520 students, 1928 are boys. How many students in the school are girls? 27. A test paper has 40 questions. Each question that is answered correctly is given 2 marks. Azar answered 37 questions correctly. How many marks did Azar score in the test?

Conceptive Worksheet 18. Add using shortcut method and verify the answers by the direct method. (I) 71 + 27 (ii) 63 + 17 + 9 (iii) 12 + 6 + 4 + 18 (iv) 756 + 347 + 251 (v) 673 + 45 + 21 + 34 (vi) 26 + 62

7 19. Subtract and verify the answers using the addition fact: (I) 2458 – 322 (ii) 1000 – 0 (iii) 9999 – 9999 (iv) 78 – 34 (v) 6736 – 369 (vi) 2273 – 1410 20. Find: (I) 32 + 46 – 12 (ii) 75 + 38 – 15 (iii) 48 – 28 + 76 + 24 (iv) 46 + 91 – 46 – 91 (v) 754 – 54 + 154 – 254 21. Fill in the blanks: (I) 10  10  _________ = 100000 (ii) 180  0 = ___________  180 (iii) 0  0 = _________ (iv) 42  16 = 40  16 + ________  16 (v) 56789  1 = ________ (vi) 7  3 + 7  2 = _________  5 22. Solve the following using shortcut method. (I) 336  5 (ii) 24  25 (iii) 78  5 (iv) 7  125 (v) 65  5 (vi) 575  25 (vii) 105  5 (viii) 325  25 23. Fill in the blanks. Dividend Quotient Divisor Remainder (i) 10 24 12 (ii) 345 12 9 (iii) 1235 102 11 (iv) 36 36 20 (v) 1648 12 137 24. A van can carry 12 pupils. How many vans would be required to carry 108 pupils? 25. Nalini has 15 tennis balls. She gives 7 tennis balls to her friend. Her father brings back another 5 tennis balls for her. How many tennis balls does she have now? 26. Madan bought 20 boxes of pens. Each box contained 15 pens. The pens were equally distributed among 25 students. How many pens did each student receive? 27. Calculate the annual salary of Raghu whose monthly salary is Rs. 2,700 28. How many children can get 6 balloons each, if there are 36 balloons in all? 29. A gardener plants 5100 mango trees in 17 rows such that each row has the same number of trees. How many trees are there in each row?

9. Rounding of Numbers 1.

Rounding a whole number means approximating it. A rounded whole number is often easier to use, understand, or remember than the precise whole number. For example, instead of trying to remember the low a state population as 2,829,252, it is much easier to remember when rounded to the nearest million: 3 million people. www.betoppers.com

6th Class Mathematics

8 To understand rounding, let’s look at examples on a number line. The whole number 36 is closer to 40 than 30, so 36 rounded to the nearest ten is 40. 20

30 36 40

50

The whole number 52 rounded to the nearest ten is 50, because 52 is closer to 50 than to 60. 52 40

50

60

70

Trying to round 25 to the nearest ten, we see that 25 is halfway between 20 and 30. It is not closer to either number. In such a case, we round to the larger ten, that is, to 30. 10

20 25 30

40

To round a whole number without using a number line, follow these steps.

Rounding Whole Numbers to a given Place Value Procedure to round off a number to a particular place value Look at the first digit to the right of the digit to be rounded off.

Digit to the right  5

Add 1 to the digit to be rounded off

Digit to the right < 5

Keep the digit to be rounded off.

Replace all the other digits to its right with ‘0’ (zeroes).

10. Estimation Strategies The estimation of numbers proves quite useful in real life. Suppose we want to purchase some clothes from the market. Before leaving home, we find a rough estimate of the amount we require to spend on the clothes and take that much money with us. But how will we estimate the amount we have to carry? Here, we will learn some estimation strategies to make our calculations easier. When we estimate a number, we should have an idea of the place value to which we are going to round off the number. In case of addition or subtraction, we first round off the numbers and then the numbers are added or subtracted.

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For example, let us estimate the sum 4369 + 263. On rounding off the numbers to the nearest thousands, we obtain 4369 rounds off to 4000 4000 263 rounds off to 0 +0 Estimated sum 4000 However, we can observe that the actual answer is 4632, which is not very close to the estimated sum which is 4000. Therefore, we should estimate the numbers to the nearest hundreds to find a more accurate answer. On rounding off the numbers to the nearest hundreds, we obtain 4369 rounds off to 4400 4400 263 rounds off to 300 +300 Estimated sum 4400 This is a better estimation as it is closer to the actual sum 4632. Similarly, we can find the product using estimation strategies. Here is a general rule for rounding off the product of two or more numbers. First round off each factor of the product to its greatest place value, and then multiply the rounded off values. For example, let us estimate the product 248 × 63. Here, 248 would be rounded off to the nearest hundred (200) and 63 to the nearest tens (60). 248 rounds off to 200 4000 63 rounds off to 60 × 60 Estimated product 12000 Therefore, the estimation of 248 × 63 is 12000. To get a more reasonable estimate, we try rounding off 63 to the nearest tens, i.e. 60, and 248 to the nearest tens, i.e. 250. We will get the answer as 250 × 60 = 15000 which is the more accurate estimation.

11. Use of Brackets To avoid confusion in calculation; such as addition, multiplication, subtraction, division; use brackets. First, calculate the values inside the brackets () into a single number and then do the operation outside. Eg: 6 (50 + 8) = 6 58 = 348 Here first we take sum of 50,8 which are inside the bracket and then resultant value is multiplied by 6.

Formative Worksheet 28. Estimate 2759 to the nearest. 29. Estimate 19378 to the nearest. 30. Estimate 165124 to the nearest.

Number System – I 31. Estimate the sum of difference. (A) 1463 + 1989 (B) 4176 + 947 (C) 730 + 9918 (D) 7961 – 341 (E) 8417 – 2499 (F) 5060 + 4068 (G) 10873 + 4759 + 3175 (H) 48931 + 4834 + 1313 32. Estimate : 3123 + 1476 + 2549 + 3173. 33. Estimate : 23,149 + 7395 + 9341 + 756. 34. Neeraj added 278, 1035 and 343 and got the sun as 1656. Use the method of estimation and find out whether the sum appears to be sensible. 35. On a particular day, a trader received Rs. 13,563 from one source and Rs. 26,587 from another. He has to pay Rs. 40,000 to someone else by the evening. By the method of estimation, find out whether he has enough money to make the payment. 36. Estimate: 6527 – 431. 37. There are 12352 workers in an industry. If 6753 workers are male, what is the estimated number of females working in the industry? 38. Estimate: (A) 87 × 313 (B) 958 × 387

9 36. Estimate the product: (A) 45 × 25 (B) 46 × 32 (C) 45 × 76 (D) 43 × 23 (E) 55 × 37 (F) 45 × 93 (G) 415 × 26 (H) 113 × 87 (I) 215 × 83 (J) 415 × 237 (K) 418 × 138 (L) 941 × 257 (M) 4965 × 102 (N) 4605 × 203 (O) 2495 × 137

12. Conversion between Hindu Arabic – Roman Numerals There are various ways of writing numerals. We use Hindu-Arabic numeral system according to which numerals are written as 1, 2, 3, 4…etc. Another way of writing numerals is Roman numeral system in which 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are written as I, II, III, IV, V, VI, VII, VIII, IX, X respectively. Roman numbers are used in clocks; they are also used in timetables to represent periods in school.

(C) 8193 × 247

Conceptive Worksheet 30. Round off 86 152 to the nearest (A) ten thousand, (B) thousand, (C) hundred. 31. Round 248,982 to the nearest hundred. 32. A new GenX automobile costs for Rs. 27,895 and a new Gen-Y automobile costs Rs. 12,329. Round each amount to the nearest thousand and estimate how much more the Gen-X costs than the Gen-Y. 33. Round each whole number to the given place. (I) 632 to the nearest ten (ii) 394 to the nearest ten (iii) 1096 to the nearest ten 34. Estimate the sum by rounding each number to the nearest ten. (I) 29 (ii) 62 35 72 42 15 + 16 + 19

Some standard symbols of Roman numerals are as follows. I V X L C D M 1 5 10 50 100 500 1000 Now, let us understand the rule to convert Roman numerals into Hindu-Arabic numerals. (i)  When a symbol of Roman numeral system is repeated, its value is added as many times as it is repeated. For example, XXX = 10 + 10 + 10 = 30 (ii)  When we write a symbol of smaller value to the right of a symbol of greater value, the value of the smaller symbol is added to the value of the greater symbol. For example, XII = 10 + 2 = 12 (iii)  If we write a symbol of smaller value to the left of a symbol of greater value, then the value of the smaller symbol is subtracted from the value of the greater symbol. For example, IX = 10 – 1 = 9

(iii) 123 138 + 147 35. Round each number to the nearest hundred to find an estimated difference.

(i)

We must remember some points while writing roman numerals. A symbol can never be repeated more than three times. For example, 40 is written as XL = 50 – 10 = 40 and not as XXXX.

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6th Class Mathematics

10 (ii) The symbols V, L, and D are never repeated. For example, we cannot write VV, LL, DD to represent numbers like 10, 100, 1000. (iii)  We can subtract I only from V and X. For example, we cannot write IL or IC to repr esent 49 or 99 respectively. 49 is represented by XLIX (XL for 40 and IX for 9). Similarly, 99 is represented by XCIX (XC for 90 and IX for 9). (iv) We can subtract X only from L, M, and C. For example, we cannot write XD to represent 490. The number is represented by CDXC (CD for 400 and XC for 90). (v)  V, L, and D can never be subtracted from any symbol. For example, we never write VXX to represent 20 or LCCC to represent 250.

13. Binary System

BINARY SYSTEM

DECIMAL SYSTEM

0

0

1

1

10

2

11

3

100

4

101

5

110

6

111

7

1000

8

1001

9

Conversion of (base two) Binary System into Decimal System

The system in which the following ten digits 0,1,2,3,4,5,6,7,8 and 9 are used to describe a number is called Decimal System. In decimal system, the digits are called ‘base digits’ . Hence this system is also called as ‘base ten system’. In a system where Digits 0 and 1 are used is called binary system or base-two system (as the base digits are only two). This system is used only for computations in computer In the decimal system, there are unit’s place, ten’s, hundred’s place and so on. Where as in the binary system there are unit’s place, two’s place, four’s place, eight’s place and so on.

Place Value Chart In the Binary System

To convert base two numerals into base-ten numerals, use place value chart of binary system. Example: To convert 1010(2) to decimal system, first consider place-value chart 1

0

1

0

23

22

21

20

1010(2)= 1 23 + 0 22 + 1 21 + 0 20 = 1 8+0+1 2+0 = 8 + 2 = 10 Write 110011(2) into decimal system The place value chart in binary system is 1

1

0

0

1

1

5

2

4

2

3

2

2

2

1

2

2

0

One’s One Hundred Sixty Twenty Eight Fours 128 27

64 26

Thirty Two’s

Sixteen’s

32 25

16 24

Eight Four’s Two’s ’s 8 23

4 22

2 21

(or) Units 1 20

Some of the numbers in binary system are 0,1,10,11,100,101,111.....and so on. They can be read as: zero, one, one-zero, onezero-zero, one-zero-one, one-one-one etc., They are written as 0(2), 1(2),10(2),11(2),100(2),101(2)..etc The following chart shows numerals in binary system along with their corresponding decimal system numbers.

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110011(2) = 1 25 + 1 24 + 0 22+ 1 21+ 1 20 = 32 + 16 + 2 + 1 = 51 Therefore 110011(2) = 51(10)

23 + 0

Conversion of Decimal System to Binary System Use division method to convert base ten numerals into base-two numerals. Here divide the given decimal system number by 2, then obtain the corresponding quotient and remainders. Continue the division until quotient gets zero. In each division, the remainder is either 0 or 1. The binary system number corresponding to given decimal system numeral is obtained by writing the remainders from bottom to top and then put 2 as base to the number.

Number System – I

11

Example: 56 can be written binary system as follows

2 56 2 28 - 0

Unit Place

2 14 - 0

Two’s Place

2

4 -0

Fourth’s Place

2

3 -1

Eight’s Place

2

1 - 1  Sixteen’s Place 0 - 1  Thirty two’s Place

Therefore 56 = 111000(2) Note: When no base is mentioned it should be treated as base 10. The number written in binary system appears to be lengthier than when it is written in the decimal system. Computers in general work with only two symbols (i.e. digits) which are

to denote

the bulb when it is switched ‘ON’, and denote when the bulb is switched ‘OFF’.

Here

to

Conceptive Worksheet 37. Write Roman numeral for: (A) 17 (B) 23 (C) 48 (D) 75 (E) 95 (F) 19 (G) 41 (H) 93 (I) 87 (J) 64 38. Write Hindu-Arabic numeral for: (A) XXIV (B) LXXI (C) XCI (D) XLIX (E) LXXV (F) XCIX (G) LXXXVI (H) LXI 39. Write Hindu-Arabic numerals for each of the following: (A) XLII (B) XLIV (C) LXVI (D) XCVIII 40. Write Roman numeral for each of the following: (A) 58 (B) 42 (C) 97 (D) 66 41. Convert the following sums into Roman numerals: (A) 3228 + 172 = 3400 (B) 2010 – 975 = 1035 (C) 32 × 58 = 1856 (D) 78 × 24 = 1872 42. Compare the following, using >, 1

4

3

2

1

0

1

2

3

4

16. Integers Numbers greater than 0 are called positive numbers. Extending the number line to the left of 0 allows us to picture negative numbers, numbers that are less than 0.

2 lies to the left of 1.  2 < 1

2.

Vertical Number Line

(A) On a vertical number line, an integer is greater than the integer below it. (B) On a vertical number line, an integer is less than the integer above it. www.betoppers.com

6th Class Mathematics

16

(D) Position Above or Below Sea level 5

(I) Sea level is taken as 0 m. (ii) If a bird flies 350 m above sea level, we write it as +350 m. (iii) If the submarine lies 150 m below sea level and write it as –150 m.

4

5 lies above 2.  5>2

3 2 1

Formative Worksheet

0

3 lies below 2.  3 < 2

1 2 3 4 5

3.

Arranging Integers in Order

1.

Number lines can be used to arrange order, integers in increasing or decreasing order. The value of integers on a horizontal number line increases from left to right and decreases from right to left.

2.

Ascending Order Values increasing Largest

Smallest 3

2

1

0

1

2

3

4

5

Descending Order Values Decreasing

IV. Writing Positive and Negative Integers to Represent Word Descriptions 1.

A positive or negative number is used to denote:

(A) An Increase or a Decrease in Value Examples: (I) Rs. 70 withdrawn is denoted by –Rs. 70. (ii) Rs.70 deposited is denoted by + Rs.70.

(B) Values more than Zero, Values less than Zero – 18 oC denotes a temperature that is 18 o C below 0 oC. (ii) +18 oC denotes a temperature that is 18 o C above 0 oC. (I)

(C) A Positive Direction or a Negative Direction (Opposite Direction) Examples: (I) –20 oC denotes an anticlockwise rotation of 20o. (ii) +20o denotes a clockwise rotation of 20o. (iii) +5 m denotes a direction 5 m to the right. (iv) –5 m denotes a direction 5 m to the left. www.betoppers.com

57. Write opposites of the following: (A) Increase in weight (B) 30 km north (C) 326 B.C. (D) Loss of Rs. 700 (E) 100 m above sea level. 58. Represent the following numbers as integers will appropriate signs. (A) An aeroplane is flying at a height two thousand metre above the ground. (B) A submarine is moving at a depth, eight hundred metre below the sea level. (C) A deposit of rupees two hundred. (D) Withdrawal of rupees seven hundred. 59. Represent the following numbers on a number line : (A) + 5 (B)-W (C) + S (D) – 1 (E) – 6. 60. Adjacent is a vertical number line, representing integers, observe it and locate the following points (A) If point D is + 8, then which point is – 8 ? (B) Is point G a negative integer or a positive integer ? (C) Write integers for points B and E. (D) Which point marked on this number line has the least value ? (E) Arrange all the points in decreasing order of value. 61. Following is the list of temperatures of five places in on a particular day of the year. Place Temperature Siachin 10°C below 0°C Shimla 2°C below 0°C Ahmedabad 30°C above 0°C Delhi 20°C above 0°C Srinagar 5°C below 0°C (A) Write the temperatures of these places in the form of integers in the blank column. (B) Following is the number line representing the temperature in desree Celsius.

Plot the name of the city against its temperature. (C) Which is the coolest place? (D) Write the names of the places whose temperatures are above 10°C.

Number System – I

17

62. In each of the following pairs, which number is to the right of the other on the number line ? (A) 2, 9 (B) –3, –8 (C) 0, –1 (D) –11, 10 (E) –6, 6 (F) 1, –100. 63. Write all the integers between the given pairs (write them in the increasing order.) (A) 0 and –7 (B) –4 and 4 (C) –8 and –15 (D) –30 and –23. 64. (A) Write four negative integers greater than –20. (B) Write four negative integers less than – 10. 65. For the following statements write True (T) or False (F). If the statement is false, correct the statement. (A) – 8 is to the right of –10 on a number line. (B) –100 is to the right of –50 on a number line. (C) Smallest negative integer is –1. (D) –26 is larger than –25. 66. Draw a number line and answer the following : (A) Which number will we reach if we move 4 numbers to the right of –2. (B) Which number will we reach if we move 5 numbers to the left of 1. (C) If we are at –8 on the number line, in which direction should we move to reach –13 ? (D) If we are at - 6 on the number line, in which direction should we move to reach –1 ?

Conceptive Worksheet 55. Use integers to repr esent the following temperatures. (A) 18 oC below freezing point (0 o(C) (B) 18 oC above freezing point (0 o(C) 56. Draw a horizontal and vertical number line to show the following integers. –15, – 10, –5, 0, 5, 10, 15 57. Which integer is smaller, – 2 or 3? 58. Which integer is greater – 3 or – 5? 59. (A) Arrange the following integers in ascending order. –5, 3, –2, 6, 2, 7, 0 (B) Arrange the following integers in descending order. 3, –1, –2, 5, –8, –5 60. Determine the largest and the smallest integers from the following integers. –12, 7, 8, 0, – 9, 5, – 10 61. Fill in the blanks in the sequence below. – 15, – 10,

,

,

, 10, 15

62. Use a positive or a negative number to denote each of the following. (A) (I) 18 m above sea level (ii) 8 m below sea level (B) (I) Profit of Rs. 188 (ii) Loss of Rs. 254 (C) (I) 15 km to the east (ii) 30 km to the west

V. Properties of Integers on Addition and Subtraction (A) If a and b are two integers then a + b = c where c is also an integer. (B) For any two integers a and b a+b=b+a Which means that if we change the order of the integers, even then their sum does not change. (C) For any three integers a, b and c (a + b) + c = a + (b + c) This means that even if we rearrange the integers their sum does not change. (D) If a is any integer then a + 0 = a This means that the sum of any integer and zero is the integer itself. Eg: – 10 + 0 = – 10 6+0=6 – 15 + 0 = – 15 (E) For every integer a (which is not zero) there is another integer – a such that a + (– a) = 0 Eg: 3 + (– 3) = 0 5 + (– 5) = 0 6 + (– 6) = 0 (F) The difference of any two integers is an integer i.e. If a and b are two integers then a – b = c, where c is also an integer. (G) In the whole numbers, 0 has no predecessor. But in integers – 1 is the predecessor of 0, –2 is the predecessor – 1 and so on. Thus if a is any integer, then a – 1 is its predecessor. (H) If a is any integer then a–0=a Like Signs Unlike Signs + (+y) = +y + (– y) = – y – (–y) = + y – (+ y) = – y

Formative Worksheet 67. Solve the following. (A) 4 + (+5) (B) 3 + (–4) 68. Evaluate –5 + (–3). 69. Simplify – 3 + 7 + (– 8).

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6th Class Mathematics

18 70. The diagram below shows a pendulum tied to a string.

Conceptive Worksheet

Table

80 cm 35 cm

71.

72.

73.

74.

When the pendulum was released from the table, it dropped to a height of 80 cm below the table. It was then pulled 35 cm up. How far is the pendulum from the table now? The temperature of a town is –14 oC at night. During the day, the temperature increases by 7 o C. What is the temperature of the town during the day? Add without using number line : (A) 11+ (-7) (B) (-13)+ ( + 18) (C) (- 10) + ( + 19) (D) ( - 250) + ( + 150) (E) (- 380) + ( - 270) (/) ( - 217) + ( - 100). Find the sum of: (A) 137 and – 354 (B) – 52 and 52 (C) – 312, 39 and 192 (D) – 50, – 200 and 300. The diagram shows a number line. x

75. 76. 77.

78.

79.

80.

81.

8

4

y

Find the value of x + y. Solve the following (A) 7 – (+6) (B) – 8 – (–3) Simplify – 8 – (+3) – (–5). A diver was diving 100 m below sea level. He went down 20 m and came up 35 m again. How far below sea level did he dive? Subtract : (A) 35 – (20) (B) 72 – (90) (C) (–15) – (–18) (D) (–20) – (13) (E) 23 – (–12) (F) (–32) – (–40) Fill in the blanks with >, < or = sign : (A) (– 3) + (– 6) _________ (– 3) – ( – 6) (B) (–21) – (–10) _________(– 31) + (– 11) (C) (45) – (– 11) _________ 57 + (– 4) (D) (–25) – (– 42) _______( – 42) – ( – 25). Fill in the blanks : (A) (–8) + __________ = 0 (B) 13 + __________ = 0 (C) 12 + (–12) = ____________ (D) (–4) + __________ = –12 (E) _____________ –15 = –10 Find the value of : (A) (– 7) – 8 – (– 25) (C) (– 13) + 32 – 8 – 1 (C) (– 7) + (– 8) + (– 90) (D) 50 – (– 40) – (– 2)

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63. Find the answers of the following additions. (A) (–11) + (–12) (B) (+10) + (+4) (C) (–32) + (–25) (D) (+23) + (+40) 64. Find the solution of the following: (A) (–7)+ (+8) (B) (–9) + (+13) (C) (+7) + (–10) (D) (+12) + (–7) 65. Find the solution of the following additions using a number line. (A) (–2) + 6 (B) (–6) + 2. 66. Find the solution of the following without using number line: (A) (+7) + (–11) (B) (–13) + (+10) (C) (–7)+ (+9) (D) (+10) + (–5). 67. Using the number line write the integer which is : (A) 3 more than 5 (B) 5 more than – 5 (C) 6 less than 2 (D) 3 less than – 2. 68. Use number line and add the following integers: (A) 9 + (–6) (B) 5 + (–ll) (C) (–1) + (–7) (D) (–5) + 10 (E) (–1) + (–2) + (–3) (F) (–2) + 8 + (–4). 69. Fill in the blanks : (I) –9 + ........... = 0 (ii) 28 + ........... = 0 (iii) 16 + (–16) = ............. (iv) –264 + .......... = –364 (v) –9 + ............ = –17 (vi) ........... + 315 = –74 70. Which of the following statements are true ? (I) –14 > –8 – (–7) (ii) –5 – (–2) > –8. (iii) The sum of two integers is always an integer.

71.

72. 73. 74. 75.

(iv) The difference of two integers is always an integer. Subtract the first integer from the second in each of the following : (I) 18,-34 (ii) –15, 25 (iii) –28, – 42 (iv) + 68, –37 (v) 219, 0 (vi) – 92, 0 (vii) –135, - 250 (viii) – 2768, – 287 (ix) 6240, - 271 (x) – 3012, 6250. Subtract –3 from 8. Are the two results the same? Find the value of –12 + (– 98) – (– 84) – (7). p and q are two integers such that p is the successor of q. Find the value of p – q. Find the value of : (I) –19 – (–16) (ii) –6 – 3 – (– 18) (iii) (3 – 5) + (3 – 5) (iv) –17 + 36 – 18 – 1 (v) 60 + (–58) – (– 4) (vi) –5 + (– 6) + (– 80).

Number System – I

Summative Worksheet 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14. 15.

Which digit in the number 568 731 has a place value of thousands? (A) 8 (B) 7 (C) 6 (D) 5 Digit 4 in the number 14 512 represents (A) 4 units (B) 4 tens (C) 4 hundreds (D) 4 thousands 20 985 rounded off to the nearest ten is (A) 20 000 (B) 20 900 (C) 20 990 (D) 21 000 Which of the following, when rounded off to the nearest thousand, is 38 000? (A) 38,400 (B) 38,500 (C) 38,600 (D) 38,700 587 315 is written as 590 000 after it has been rounded off to (A) the nearest ten (B) the nearest hundred (C) the nearest thousand (D) the nearest ten thousand 1,384 + 5,580 + 47,218 = (A) 54,182 (B) 54,178 (C) 51,184 (D) 51,178 The sum of which of the following is the smallest? (A) 923 + 456 (B) 701 + 632 (C) 602 + 788 (D) 513 + 998 83,102 = 80,000 + 3,000 + ______ + ____ + 2 (A) 1000, 100 (B) 1000, 10 (C) 100, 10 (D) 100, 0 A basket of apples weigh 500 g. If Kaveri bought 7 baskets of apples, how much would they weigh? (A) 3,500 kg (B) 350 kg (C) 35 kg (D) 3.5 kg Universal Tuition Centre has 1,070 students. Of these, 598 are boys. How many students in the centre are girls? (A) 1,668 (B) 488 (C) 472 (D) 462 2,493 – 276 = (A) 2,769 (B) 2,627 (C) 2,227 (D) 2,217 371 503 – 281 498 = (A) 91,005 (B) 90,905 (C) 90,095 (D) 90,005 Which of the following has the largest value? (A) 32,457 + 42,864 (B) 33,247 + 41,684 (C) 32,574 + 42,684 (D) 33,427 + 41,864 209 – 21 – 69 = (A) 119 (B) 109 (C) 99 (D) 89 Shyam earns Rs. 3 500. If he spent Rs. 1 249 on buying a handphone, what is the balance of his salary? (A) Rs. 2 751 (B) Rs. 2 251 (C) Rs. 2 151 (D) Rs. 2 059

19 16. The integer – 18 is read as (A) minus 18 (B) negative 18 (C) subtract 18 (D) negative 18 units 17. A glass of water is at its freezing point of 0 oC. When salt is added, its temperature drops to 5 o C below zero. What is the temperature of the water? (A) 5 oC (B) +5 oC o (C) –5 C (D) 5 oC below zero 18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

28 21

S 7

0

T

The integers represented by S and T on the above number line are: S T S T (A) – 14 7 (B) –14 2 (C) – 8 14 (D) – 14 14 Which of the following is incorrect? (A) 4 > 0 (B) 5 < – 3 (C) 7 > – 3 (D) 8 < 9 Which set of integers below is not in descending order? (A) 0, 1, 2, 3, 4 (B) 3, 2, 1, 0, – 1 (C) 7, 6, 5, 4, 3 (D) –5, –6, –7, –8, –9 Which of the following integers has the greatest value? (A) –11 (B) – 10 (C) – 9 (D) – 8 If +8 means 8 units forward, then – 8 means 8 units (A) forward (B) to the right (C) backward (D) to the left If a profit of Rs. 2180 is written as +Rs. 2180, then a loss of Rs. 2180 should be written as (A) Rs.0 (B) Rs. 180 (C) Rs. –2180 (D) –Rs. 2180 Draw a number line and represent each of the following on it : (I) 9 + (– 6) (ii) (– 3) + 7 (iii) 8 + (– 8) (iv) 3 + (– 2) + (– 4) (v) (–1) + (– 2) + (– 3) (vi) 5 + (– 2) + (– 6). Add the integers in each of the following : (I) –365 –87 (ii) 1001, – 13 (iii) – 3057, 199 (iv) – 36, + 1027 (v) – 389, – 1032 (vi) – 36, 100 (vii) – 312, 39, 192 (viii) – 51, – 203, 36, – 28 (ix) 3002, – 888 (x) – 18, 25, – 37. Find the sum : (i) (–7) + (–9)+ 4 + (–10) (ii) 37 + (– 23) –f (– 65) + (– 8) (iii) (–147) + 79 + (–265) + (–39) (iv) 056 + (–798) + (–38) + 56. Let us invent an operation * for integers such that for two integers a and b. a * b = a + b + (–2a) Determine (i) (– 4) * 5 (ii) 5* (– 2). Which of the following statements are true ? (i) Zero is a negative integer. (ii) 0 + 0 = 0 (iii) 0 is the smallest integer. www.betoppers.com

6th Class Mathematics

20

29. 30.

31. 32. 33.

(iv) The successor - 61 is - 62. (v) Sum of 3 different integers is always zero. (vi) The predecessor of –200 is –201. The sum of two integers is –337. If one of them is 250, find the other. A place is 37 m above sea level and another is 35 m, below sea level. What is the difference of level between the two places ? The sum of two integers is - 639. If one of them is 529, find the other. Find the difference of the sum of odd numbers and sum of even numbers between 10 and 20. Subtract the sum of –8 and –28 from the sum of –13 and 31.

HOTS Worksheet 1.

Add the following : The smallest four-digit whole number, the greatest three-digit whole number and the smallest five-digit whole number. 2. Write all the possible three-digit numbers using the digits 6,8,0 and add them. 3. Write and add three consecutive numbers from 96,778. 4. There are 12,000 men, 11,789 women and 1,700 children in a village. Find the population of the village. 5. Kishore had Rs. 32,840. He deposited Rs. 17000 in a savings bank account and Rs. 2,500 in a fixed deposit account in a bank. How much money was left with him? 6. The cost of a wet-grinder is Rs. 3,755. The cost of a fruit-juicer is Rs. 1,250 less than that of the wet-grinder. Find the total cost of the wet-grinder and the fruit juicer. 7. A 29-inch colour TV set costs Rs.32,565 and a 21 inch model costs Rs. 18,675. How much more does the bigger TV set cost? 8. Do as directed: (i) Divide 120 by the product of 4 and 3. (ii) Add the greatest two digit-number to the difference of 14 and 3. (iii) Subtract product of 61 and 17 from 1,685 (iv) Add the quotient of 750 and 15 to the difference of 75 and 33. (v) Multiply 715 by 20 and divide by 25 9. 2,500 metres of cotton cloth was made into 20 rolls of 30 m each and 25 rolls of 20 m each. Find the length of the remaining cloth in metres and how many rolls of 70 m can be made with the remaining cloth. 10. In a lending library the number of books borrowed per day in one week are 101, 243, 316, 443, 890, 1252 and 1535 respectively. How many books were borrowed in one week? If the books are lent out at Rs. 10 each, find the amount of money collected by the lending library in the week. www.betoppers.com

11. The first Class fare from Chennai to Coimbatore is Rs. 557 and the sleeper class fare is Rs. 165. How much would a family of 4 members save by choosing to travel by sleeper class? 12. A day consists of 24 hours. How many hours will there be in the year 2005 and 2008 and find the difference between them? 13. The stock market average fell 317 points in one day. Represent this quantity by an integer. 14. The lowest elevation in India is found at Nilgir Valley at an elevation of 282 feet below sea level. Represent this elevation with an integer. 15. The temperature on one January day in Delhi was – 10 degrees Celsius. Tell whether this temperature is cooler or warmer than – 5 degrees Celsius. 16. The temperature at 4 P.M. on February 2nd was – 10o Celsius. By 11 P.M. the temperature had risen 12 degrees. Find the temperature at 11P.M. 17. Scores in golf can be positive or negative integers. For example, a score of 3 over par can be represented by + 3 and a score of 5 under par can be represented by – 5. If a Couple had scores of 3 over par, 6 under par, and 7 under par for three games of golf, what was the total score? 18. Suppose a deep-sea diver dives from the surface to 165 feet below the surface. He then dives down 16 more feet. Use positive and negative numbers to represent this situation. Then find the diver’s present depth. 19. Suppose a diver dives from the surface to 248 meters below the surface and then swims up 6 meters, down 17 meters, down another 24 meters, and then up 23 meters. Use positive and negative numbers to represent this situation. Then find the diver ’s depth after these movements. 20. Arun has Rs. 125 in his saving account. He writes a cheque for Rs. 117, makes a deposit of Rs. 45, and then write another cheque for Rs. 69. Find the amount left in his account. (Write the amount as an integer). 21. In a test, it is possible to have a negative score. If Krishna’s score is 15, what is his new score if he loses 20 points? 22. The temperature on a February morning is – 6o Celsius at 6 A.M. If the temperature drops 3 degrees by 7 A.M., rises 4 degrees between 7 A.M. and 8 A.M., and then drops 7 degrees between 8 A.M. and 9 A.M,. find the temperature at 9 A.M. 23. A mountain peak in Himalayas has an elevation of 14,393 feet above sea level. A deep-sea trench in the Pacific Ocean has an elevation of 12,456 feet below sea level. Find the difference in elevation between these two points.

Number System – I

IIT JEE Worksheet I.

Single Correct Answer Type

1.

2,493 – 276 = (A) 2,769 (B) 2,627 (C) 2,227 (D) 2,217 371 503 – 281 498 = (A) 91,005 (B) 90,905 (C) 90,095 (D) 90,005 Which of the following has the largest value? (A) 32,457 + 42,864 (B) 33,247 + 41,684 (C) 32,574 + 42,684 (D) 33,427 + 41,864 209 – 21 – 69 = (A) 119 (B) 109 (C) 99 (D) 89 Shyam earns Rs. 3 500. If he spent Rs. 1 249 on buying a hand phone, what is the balance of his salary? (A) Rs. 2 751 (B) Rs. 2 251 (C) Rs. 2 151 (D) Rs. 2 059 Smriti bought 19 boxes of sweets. Each box contains 228 sweets. How many sweets would be left with her after giving 519 sweets to friends? (A) 766 (B) 3,813 (C) 4,332 (D) 4,851 4,668  4 = (A) 1,267 (B) 1,167 (C) 1,087 (D) 1,012 8,158  32 = (A) 254 remainder 30 (B) 242 remainder 30 (C) 204 remainder 15 (D) 202 remainder 15 The company divided 15,484 workers equally into 98 groups. How many workers were there in each group? (A) 178 (B) 168 (C) 158 (D) 148 Which of the following is true? (A) 130  0 = 0 (B) 150  1 = 50 (C) 0 × 130 = 0 (D) 150 × 1 = 151 If –30 + (–y) + 20 = 0, then the value of y is (A) 20 (B) 10 (C) –10 (D) –20 –40 – 32 + 40 = (A) – 80 (B) – 32 (C) 32 (D) 80 0 – (–3) – (–4) = (A) – 7 (B) – 1 (C) 1 (D) 7 – 8 – (–3) – (–5) = (A) 16 (B) 6 (C) 0 (D) – 16 – 10 – (–2) – 3 = (A) – 10 (B) – 11 (C) – 12 (D) – 15 In a quiz, each student is required to answer 40 questions. 5 marks are given for every correct answer and 3 marks are deducted for every wrong answer. If Mahesh answered 35 questions correctly and Suresh answered 32 questions correctly, what is the difference in the total marks obtained by them? (A) 15 (B) 24 (C) 160 (D) 175

2. 3.

4. 5.

6.

7. 8.

9.

10.

11. 12. 13. 14. 15. 16.

21 17. If the temperature of City A is –20 oC and the temperature of City B is 10 oC, the difference in temperature between the two cities is (A) –30 oC (B) – 10 oC (C) 10 oC (D) 30 oC 18. – 40 + (–5) + (–3) = (A) –48 (B) – 32 (C) 32 (D) 48 19. The diagram shows a number line. x 8

0

y

The value of x + y is (A) – 32 (B) – 8 (C) 8 (D) 32 20. A lady parked her car on the 6th floor and took a lift up 17 floors to the Finance Department. She then went down 9 floors to the Tax Department. On which floor would you find the lady? (A) 14th (B) 15th (C) 23rd (D) 26th

II.

Multiple Correct Answer Type

21. A set of whole numbers satisfy ___________ property/properties under multiplicative. (A) closed (B) commutative (C) associative (D) none 22. For any three elements a,b,c  N a  (b – c)  (A) ac – bc (B) bc – ac (C) ab – ac (D) ab – bc 23. Which of the following would give the value zero? (A) 30  0 (B) 30  0 (C) 0  30 (D) 0  0 24. X can be subtracted from: (A) D (B) L (C) M (D) C 25. Which of the following sequence of operations are true, to simplify the combination of operations that involve addition (A) , Subtraction (S), Multiplication (M) and Division (D). (A) M – A – S (B) M – A – D (C) D – M – A (D) S – A – M 26. Integer greater than –151: (A) –141 (B) –120 (C) –251 (D) –111 27. –4  –3  5 is equal to: (A) –4  (–3  5) (B) –4  (–3 + 5) (C) (–4  3)  (–4  5) (D) (–4  –3)  5 28. (–2)  (–2)  (–2)  (–2)  (–2) is: (A) +32 (B) (–2)5 (C) (–32) (D) (2)5 29. Compare: (–15) and (–16) (A) (–15) = (–16) (B) (–15) > (–16) (C) (–16) < (–15) (D) none of these 30. Say which of the following statements is true? (A) (–a) is the additive inverse of (+a) (B) for any integer a, a  1  1  a (C) (–27) is a perfect cube because (–27) = (–3)3 (D) ‘0’ is a positive integer

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6th Class Mathematics

22

III. Paragraph Type

Paragraph – III

Paragraph – I

32.

33. 34.

Paragraph – II

35.

36.

37.

38.

39.

The following table shows the top 10 leading advertisers in 2008 and the amount of money spent in that year on ads. Procter & Gamble Rs. 1,46,49,94,000 General Motors Rs. 1,39,88,06,000 Philips Rs. 1,30,76,61,000 Ford Motor Rs. 92,02,64,000 Reliance Rs. 75,86,84,000 AT & T Rs. 70,04,29,000 Pepsi Rs. 66,96,86,000 Toyato Motor Rs. 53,61,23,000 Reebok Rs. 50,76,62,000 TATA Rs. 50,36,21,000 Based on this information answer the questions given below. Which companies spent more than Rs. 1 billion on ads? (A) Proctor & Gamble (B) General Motors (C) Philips (D) All Which companies spent less than Rs. 700 million on ads? (A) Pepsi (B) Toyoto Motor (C) Reebok (D) All How much more money did Philips spend on ads than TATA? (A) 804040000 (B) 8040400 (C) 404040000 (D) 4040400 How much more money did AT &T spend on ads than Toyata Motor? (A) 804040000 (B) 164306000 (C) 1643060 (D) 789546000 Find the total amount of money spent by these ten companies on ads. (A) 6767930000 (B) 9767930000 (C) 8767930000 (D) 8787870000

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Game 1 Game 2 Game 3

Game 4

Team 1 –2 – 13 20 2 Team 2 5 11 –7 –3 Based on this information answer the questions given below. 40. If the winner is the team with the greater score, find the winning team. (A) Team 1 (B) Team 2 (C) both (D) none 41. Find the total score of team 1 after 4 games. (A) 4 (B) 5 (C) 6 (D) 7 42. What is the least score scored by the by the teams. (A) –2 (B) –3 (C) –7 (D) –13

Paragraph – IV The following bar graph shows heights of selected lakes. Feet above or below Sea Level

31.

Johny has 36 blue marbles and 54 red marbles. He wants to put an equal number of blue and red marbles into some boxes. Based on this information answer the questions given below. How many marbles are there with Johny in all? (A) 36 (B) 54 (C) 90 (D) 180 By what number are blue marble less than red marbles? (A) 18 (B) 16 (C) 14 (D) 12 How many boxes does Johny need atmost? (A) 3 (B) 6 (C) 9 (D) 18 What number of groups can be made of 54 red marbles if each group consists of 9 marbles? (A) 3 (B) 6 (C) 9 (D) 18

In some card games, it is possible to have positive and negative scores. The table below shows the scores for two teams playing a series of four card games.

600

600 512

500 400 300

245

200

144

100 0

92

0

C

D

52

100 A

43. 44. 45. 46.

B

E

F

G

Based on this information answer the questions given below. Find the difference in elevation for the lakes listed. Lake A and Lake F. (A) 40 ft (B) 144 ft (C) 604 ft (D) 652 ft Lake E and Lake C. (A) 40 ft (B) 144 ft (C) 604 ft (D) 652 ft Lake G and Lake D. (A) 40 ft (B) 144 ft (C) 604 ft (D) 652 ft Lake F and Lake C. (A) 40 ft (B) 144 ft (C) 604 ft (D) 652 ft

Number System – I

23

IV. Integer Type 47. ×

267 54

10 68 1035 11418

48. 49. 50 51. 52. 53. 54.

In the above calculation, the answer 11,418 is incorrect due to a mistake in one of the digits. The incorrect digit is __________. 244 + (8 × 4) – 318  a = 223. The value of a is ___________. 100 – 7 × 1 + 5 = 100 – (7 × 1) + a. The value of a is ________. Which digit in the number 568 731 has a place value of thousands? In the number 14 512 represents, the number in thousands place is _________. Fill in the missing integer: 6 + =0 6 – (–3) = _______ The temperature at 4 P.M. on February 2nd was – 10o Celsius. By 11 P.M. the temperature had risen 12 degrees. Find the temperature in °C at 11 P.M.

V. Matrix Matching (Match the following) 55.

Column - I Column - II (A) The difference of the place values of two 4’s in 4324(p) 0 (B) The difference between the face values of two 4’s in 24348 is (q) 1114 (C) The Indo-Arabic numeral corresponding to MCXIV (r) 2224 (D) 12345678 x (5-5) (s) 3996 (t) 0/12456789 56. Column - I Column - II (A) (25) - (-25) (p) -6 (B) -30 + 50 (q) -18oC o (C) 18 C below freezing point 0oC) (r) 1114 (D) -8-(+3)-(-5) (s) -80 (t) 50 (u) +18oC (v) 20 (w) +16

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6th Class Mathematics

Number System-II

25

B y th e e n d o f th i s c h a p te r , yo u w i l l un de r stan d I. • • • • • • •

Factors & Multiples Multiples Factors Even and Odd Numbers Prime and Composite Numbers Divisibility H.C.F for G.C.D L.C.M.

II. • • • • • • • •

Fractions & Decimals Fractions Classification of Fractions Interconversion of Fractions Comparison of Fractions Decimals and Decimal places Inter-conversion of decimal and fractions Comparison of Decimals Fundamental Operations on Decimals

III. • • • •

Squares and Square Roots Squares of numbers Perfect squares Square roots Finding square roots

Chapter -2

Number System-II

Learning Outcomes

Factors & Multiples 1. Multiples The numbers obtained by multiplying the given number with a natural number are called multiples of a given number. Example: 2 1 = 2, 2 2 = 4, 2 3 = 6, 2 4 = 8 and so on. Therefore 2, 4, 6, 8....etc are called multiples of 2.

numbers and set of natural numbers are infinite. (vii) For two or more numbers, the multiples which are common to both numbers are called common multiples. i.e., for any two numbers a,b; a b = ab. The product of ‘ab’ is common multiple of both ‘a’ and ‘b’ Example: 2 × 3 = 6; 6 is common multiple of 2 and 3 (since the multiples of 2 are 2, 4, 6, 8 etc., and the multiples of 3 are 3, 6, 9, 12 etc.,)

3,6,9,12,15......etc are called multiples of 3. Note: (i)

Every number is a multiple of itself.

(ii) Every number is multiple of one. Since 1 1 = 1, 1 2 = 2, 1 3 = 3.....etc. (iii) There are infinite number of multiples for a natural number. (iv) When a number divides another number exactly, then dividend is called a multiple of the divisor. For example, 16 is exactly divisible by 4. Here 16 is dividend and 4 is divisor. 16 is multiple of 4 (since 4×4 = 16) (v) When two or more numbers are multiplied, then the product is a multiple of each number. Example:2 3 4 = 24; 24 is multiple of 2, 3, 4 4

5 = 20; 20 is multiple of 4, 5

(vi) For every number, number itself is a least multiple. Example: 2,4,6,8....are multiples of 2, among these multiples 2 is a least multiple of 2. (vii) The greatest multiple of a number does not exist, since the multiples of a number are products of number with natural

2. Factors For any two numbers a,b; if ‘ab’ is the product of a, b then each of a, b are called factors of product ab. Example:2

3 = 6, where 2, 3 are factors of

product 6. When a number divides another number exactly (it means it leaves remainder 0); then the divisor is called a factor of the dividend. Example: 2 divides 18 and leaves a remainder 0. i.e., 2 divides 18 exactly. Therefore 2 is a factor of 18, where 2 is divisor and 18 is dividend. When two or more numbers are multiplied then each number is a factor of the product. Example: 2 × 3 × 4 = 24 where 2, 3, 4 are called factors of 24 1 × 12 = 12, 2 6 = 12, 3 × 4 = 12. Therefore 1, 2, 3, 4, 6, 12 are factors of 12 Note: (i)

We denote the set of all factors of ‘n’ by F(n) or Fn. www.betoppers.com

6th Class Mathematics

26 Example: F(12) = F12 = {1,2,3,4,6,12} (ii) The smallest factor of any number is 1 and the largest factor is the number itself. Example:1, 2, 3, 4, 6, 12 are factors of 12. Among these factors, the smallest factor is 1 and the greatest (or larger) factor is 12. (iii) 1 is a factor of every number and is a smallest factor (iv) Every number is a factor of itself and is largest factor. (v) The factor of a number is less than or equal to that number. Example: The factors 12 are less than are equal to 12 (since 1 < 12, 2 < 12, 3 < 12, 4 < 12, 6 < 12, 12 12)

3. Even and Odd Numbers I.

Even Numbers All the whole numbers which are exactly divisible by 2 are called even numbers. Example: 0, 2, 4, 6, 8,10, 12, .................

are even numbers (as each is exactly divisible by 2). There are infinite even numbers. II.

Odd Numbers All the whole numbers which are not exactly divisible by 2 are called odd numbers. Example:1, 3, 5, 7, 9, 11, ............. are

odd numbers (as each is not exactly divisible by 2). There are infinite odd numbers.

4. Prime and Composite Numbers I.

Prime Numbers Natural numbers which have exactly two distinct factors (1 and the number itself) are called Prime numbers. The natural numbers 2, 3, 5, 7, 11, 13, 17, 19, ...... are some prime numbers. (A) There are infinite prime numbers. (B) 2 is the only prime number which is even. All other prime numbers are odd. (C) Every odd number is not prime. For example 9, 15, 21, ....... are odd numbers, but not prime numbers.

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II.

Composite Numbers Natural numbers which have more than two factors (at least one more factor other than 1 and the number itself) are called composite numbers. The natural numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, ..... are some composite numbers. (A) Ther e are infinite composite numbers. (B) 1 is neither prime nor composite. It is called unity. (C) All even numbers (except 1) is either prime or composite. (D) All even numbers (except 2) are composite numbers. e)

Some odd numbers are composite and some are prime.

III. Methods of finding Prime Numbers Method 1: To find prime numbers between 1 and 100 Eratosthenes, a Greek mathematician devised a simple method. Procedure: Write down the numbers from 1 to 100 in rows of ten. Cross out 1. Encircle 2 and cross out all the multiples of 2 i.e 4, 6, 8....etc Encircle 3 and cross out all the multiples of 3. i.e 6, 9, 12, 15, 18, 21 etc. Encircle 5 and cross out all the multiples of 5 i.e 10, 15, 20, 25....etc Encircle 7 and cross out all the multiples of 7 i.e. 14, 21, 28....etc Now encircle each one of remaining numbers and cros sout all the multiples of all of them if exist. Continue this process till all the numbers in the list are either encir cled or crossed out. Now the encircled numbers are the Prime Numbers and all the crossed out numbers are Composite Numbers. Thus we get 25 prime numbers between 1 and 100. The tabular arrangement of numbers obtained below is named as sieve of eratosthenes.

Number System-II

27 2

3

4

5

6

7

8

9

10

11

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17

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40

2

3

5

7

41

42

43

44

45

46

47

48

49

50

11

13

17

19

23

29

31

37

41

43

47

53

59

61

67

71

73

79

83

89

97

101 103 107

51

52

53

54

55

56

57

58

59

60

61

62

63

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66

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68

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80

81

82

83

84

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86

87

88

89

90

91

92

93

94

95

96

97

98

99 100

Prime numbers

109 113

101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

(ix) The number of prime numbers between 81 to 90 is 2

Method 2: To test a number ‘n’ whether it is prime or not, take the square root of ‘n’ and consider as if it is a natural number; otherwise just increase the square root of it to the next natural number. Then divide the given number by all the prime numbers less than the square root obtained for that number. If the number is divisible by any of these prime numbers then it is not a prime number else it is a prime number. Example: Check whether 241 is prime.

(x) The number of prime numbers between 91 to 100 is 1 (xi) The number of prime numbers between 101 to 110 is 4 (xii) The number of prime numbers between 111 to 120 is 1 The Number of Prime Numbers between 1 to 100 are 25

IV. Co-Prime Numbers

When we take the square root of 241 it is approximated to 16, so we consider it 16. Now divide 241 by all the prime numbers below 16 are 2, 3, 5, 11, 13.Since 241 is not divisible by any one of the prime numbers below 16, it is a prime number. Note: (i) The number of prime numbers between 1 to 10 is 4 (ii) The number of prime numbers between 11 to 20 is 4 (iii) The number of prime numbers between 21 to 30 is 2 (iv) The number of prime numbers between 31 to 40 is 2 (v) The number of prime numbers between 41 to 50 is 3 (vi) The number of prime numbers between 51 to 60 is 2 (vii) The number of prime numbers between 61 to 70 is 2 (viii) The number of prime numbers between 71 to 80 is 3

Two natural numbers are said to be coprime numbers (or relatively prime numbers) if they have only 1 as common factor. Example:2, 3 ; 2, 5 ; 3, 10; 15, 16 ; .....

are co-prime numbers. V.

Twin Primes Two prime numbers which have only one composite number between them are called twin primes. Example:3, 5, ; 5, 7 ; 11, 13; 17, 19 ;

29, 31; 41, 43; 59, 61 ; 71, 73; ...... are twin primes. VI. Perfect Number If the sum of all the factors of a number is two times the number, then the number is called a perfect number. Example: Factors of 6 are 1, 2, 3 and 6.

Sum of all the factors = 1 + 2 + 3 + 6 = 12 = 2(6)  6 is a perfect number..

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28

5. Tests for divisibility of Natural Numbers I.

II.

Formative Worksheet

Divisibility by 2

1.

Find a perfect number between 25 and 30.

A natural number is divisible by 2 if its units digit is 0, 2, 4, 6 or 8 (i.e. unit’s digit is divisible by 2)

2.

Which of the numbers are divisible by 3? (A) 932105

3.

Divisibility by 3 A natural number is divisible by 3 if the sum of its digits is divisible by 3.

III. Divisibility by 4 A natural number is divisible by 4 if the number formed by its last two (i.e. ten’s and unit’s) digits is divisible by 4.

IV. Divisibility by 5

6.

V.

Divisibility by 6

7.

VI. Divisibility by 8 A natural number is divisible by 8 if the number formed by its last three (Hundred’s, ten’s and unit’s) digits is divisible by 8.

VII. Divisibility by 9

VIII. Divisibility by 10

(B) 8179320

(B) 2455439

Conceptive Worksheet 1.

A natural number is divisible by 10 if its last (unit’s) digit is 0.

IX. Divisibility by 11 A natural number is divisible by 11 if the difference of the sums of digits at the alternative places (starting from unit’s place) is divisible by 11.

Work out the following sums and write your conclusions below for a, b, c, d: (i) 37 – 13

(ii) 28 – 17

(iii) 43 – 14

(iv) 56 – 38

(A) Odd number – Odd number = _______ (B) Even number – Odd number = _______ (C) Odd number – Even number = _______

Some Observations (A) All numbers divisible by 2 are even.

(B) 4204561

Which of the following numbers are divisible by 11? (A) 2221582

A natural number is divisible by 9 if the sum of its digits is divisible by 9.

(B) 5719842

Which of the following numbers are divisible by 10? (A) 500505

9.

(B) 98274

Which of the following numbers are divisible by 9? (A) 634680

8.

(C) 13790

Which of the following numbers are divisible by 8? (A) 987048

A natural number is divisible by 6 if it is divisible by 2 as well as by 3.

(B) 50502

Which of the following numbers are divisible by 6? (A) 24056

A natural number is divisible by 5 if its last (unit’s) digit is 0 or 5.

(B) 987542

Which of the numbers are divisible by 5? (A) 460765

5.

(C) 262242

Which of the following numbers are divisible by 4? (A) 75020

4.

(B) 4980204

(D) Even number – Even number = _______ 2.

(B) Numbers divisible 3 may be even or odd.

Find the following products and write your conclusions below:

(C) Numbers which are divisible by 2 and 3 are divisible by 6.

(i) 37 × 7

(ii) 28 × 9

(iii) 69 × 6

(iv) 52 × 4

(D) Numbers which are divisible by 4 are also divisible by 2.

(A) Product of two odd numbers = ______ (B) Product of an odd number and an even number = _______

(E) Numbers which are divisible by 8 are also divisible by 2 and 4. (F) Numbers which are divisible by 9 are also divisible by 3. (G) Numbers which are divisible by 10 are also divisible by 2 and 5. www.betoppers.com

(C) Product of two even numbers = _______ 3.

Find the difference between the smallest and largest prime numbers between 6 and 18. (A) 4

(B) 6

(C) 8

(D) 10

Number System-II 4. The sum of all the prime factors of m is 9. Find the number m. (A) 15

(B) 20

(C) 28

(D) 63

5. Which of the following is even - 2, 2, 3 ? _________________________________________________________________________________

6. Common Factors Common Multiples II.

and

Common Factors Observe the factors of some numbers taken in pairs. (A) What are the factors of 4 and 18? The factors of 4 are 1, 2 and 4. The factors of 18 are 1, 2, 3, 6, 9 and 18. The numbers 1 and 2 are the factors of both 4 and 18. They are the common factors of 4 and 18.

II.

Common Multiples What are the multiples of 4 and 6? The multiples of 4 are 4, 8, 12, 16, 20, 24, ... (write a few more)

29

7. Highest Common Factor (H.C.F) or Greatest Common Divisor (G.C.D) H.C.F. of two or more natural numbers is the largest (or highest) common factor (or divisor) of the given numbers. Thus, H.C.F. is equal to the greatest element of the set of common factors (or divisors) of the given numbers. Therefore it is also called Greatest Common Divisor (G.C.D.) Note:H.C.F. of two or more numbers divides

each number completely. Methods of finding H.C.F. I.

Continuous Division Method In continuous division method, we divide the larger number by the smaller number and get a remainder. Then we divide the first divisor by the remainder and get a new remainder. Continue this process till the last remainder is zero. The last divisor in this process is the H.C.F. of the given two numbers. Example: Find the HCF of 180 and 324

The multiples of 6 are 6, 12, 18, 24, 30, 36, ... (write a few more)

180

144

36

Here we observe that when we factorise 36 = 2×2×3×3, then all its factors are primes. Such factorisation of a number is called prime factorisation. Thus if a natural number is expressed as the product of prime numbers, then the factorisation of the number is called its prime factorisation.

144

4

144

Can you write a few more?

We are familiar with the terms factors and multiples. So let us factorise 36 in different ways as: 36 = 1× 36, 36 = 2×18, 36= 3×12, 36 = 4 × 9, 36 = 6×6, 36 = 2×2×9, 36 = 2×6×3, 36 = 3×3×4, 36 = 2×2×3×3

180 1 144

We observe that 12, 24, 36, ... are multiples of both 4 and 6.

III. Prime Factorisation

1

180

Out of these, are there any numbers which occur in both the lists?

They are called the common multiples of 4 and 6.

324

0

 HCF of 180 and 324 is 36.

To find the H.C.F. of three or more numbers, we proceed as: i)

Find the H.C.F. of any two given numbers.

ii)

Find the H.C.F. of the third number with the H.C.F. of step (i).

iii)

H.C.F. obtained in step (ii) is the required H.C.F. of the three given numbers.

iv)

For more numbers, we continue this process.

Example: Find the HCF of 24, 32 and 44. First consider 24 and 32.

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6th Class Mathematics

30 24

32

I.

1

24 8

24

In division method, we divide the given numbers by a common divisor of at least two of the given numbers and write the numbers as they are which are not divisible by this common divisor. We continue this process till no two numbers have a common factor. Then L.C.M. of the given numbers is the product of the divisors and the remaining numbers.

3

24 0

Now, consider 8 and 44 8

44

5

40 4

8

Methods of finding L.C.M. Division Method

2

Example: Find the LCM of 336 and 420

8 0

II.

2 336, 420

 HCF of 24, 32 and 44 = 4

2 168, 210

Prime Factorisation Method

3 84, 105 7 28, 35

In prime factorisation method, we find the prime factorisation of each of the given numbers. Then H.C.F. is equal to the product of all the different common prime factors of the given numbers using each common factor the least number of times it appears in the prime factorisation of all of the given numbers.

4, 5

LCM

= 1680

II.

2 216 2 108 2 54

2 30

3 27

3 15 5

60 = 2  2  3  5

3

9

3

3

216 = 2  2  2  3  3  3

8. Least Common (L.C.M)

Multiple

L.C.M. of two natural numbers is equal to the smallest natural number which is a multiple of both the numbers. Thus, L.C.M. is equal to the smallest element of the set of common multiples of the given natural numbers. Example: Find the LCM of 6 and 12. Multiples of 6 = {6, 12, 18, 24, 30, 36,........} Multiples of 12 = {12, 24, 36, ..............} Set of common multiples 6 and 12 ={12, 36, ...} The smallest element of this set = 12  LCM = 12

Relation between H.C.F. and L.C.M. of two Natural Numbers Product of H.C.F.and L.C.M. of two natural numbers is equal to the product of the two numbers. H.C.F.  L.C.M. = Product of the two numbers.

Example: Find the HCF of 60 and 216.

2 60

= 2 2 3 7 4 5

Formative Worksheet 1.

Is 180 a common multiple of 3, 4 and 5?

2.

Identify the common factors of 6,8 and 12.

3.

Find the common factors of 10 and 15.

4.

Express the following numbers as the product of primes. (A) 90

(B) 675

(C) 1089

5.

Find the H.C.F. of 72, 192 and 324.

6.

What is the HCF of 6 and 12?

7.

Find the GCD of 36 and 48

8.

Find the L.C.M. of 6 and 9 by division method.

9.

Find the L.C.M. of 6 and 9 by prime factorisation method

10. If the product of two numbers is 336 and their H.C.F. is 4, find their L.C.M. 11. Find the LCM of 18, 24, 60 and 120 by division method. 12. Find the GCD of 136 and 120. 13. Find the GCD of the 459, 357, 306

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Number System-II

Conceptive Worksheet 1.

Which of the following is a multiple of 24 ?

2.

The prime factors of 96 are ?

3.

The number of divisors (factors) of 136 is ?

4.

Is 156 is a common multiple of 3, 13 ?

5.

The sum of all the prime factors of a number is 15, then the number is ?

6.

Give an example of twin primes ?

7.

Is 4 a common factor of 36, 44 and 100?

8.

Find the factor of 20

9.

Determine the HCF of 14 and 20.

7.

31 Find the greatest number that can divide 72, 108 and 144 without remainder.

8.

Three drums contain 144, 240 and 180 litres of oil respectively. Find the greatest capacity of a bucket which can empty out each drum with an exact number of filling.

9.

Find the LCM by prime factorisation: (A) 90, 126, 135

(B) 15, 18, 36, 45

(C) 96, 588

(D) 13, 18, 39, 90

(E) 28, 49, 91 105 10. Find the LCM by division method:

10. Find the H.C.F. of 16 and 24.

(A) 60, 120, 320

(B) 36, 144, 100

(C) 20, 25, 36, 72

(D) 55, 99, 132, 121

(E) 115, 125, 130

11. Find the L.C.M. of 6 and 9. 12. Prove that for the numbers 24 and 40, H.C.F.  L.C.M. = Product of the two numbers 24 and 40

11. Find the sum of all the common factors of 10 and 25. 12. Find the highest common factor of 35 and 56. 13. List all the factors of 27 and 52.

13. Find the GCD of 30, 75 and 135

Summative Worksheet 1.

2.

HOTS Worksheet 1.

Find the factors of each of the following numbers: 18, 32, 48, 100, 144

(B) Determine the lowest common multiple of 6 and 9.

Find the common factors of the following numbers:

(C) List all the factors of 81. (D) List all the prime factors of 72.

(A) 25, 45, 70; (B) 27, 36, 81; (C) 64, 16, 48 3.

4.

5.

6.

(A) What is the lowest common multiple of 2, 5 and 11?

(E) Determine the common factors of 12 and 27.

Using the factor tree method find the prime factors of: (A) 126

(B) 240

(D) 315

(E) 468

(C) 144

2.

(B) Find the lowest common multiple of 6, 11 and 18.

Resolve into prime factors using the division method: (A) 595

(B) 704

(D) 2180

(E) 3432

(C) Find the lowest common multiple of 8, 12 and 20.

(C) 612

(D) Find the largest number which divides both 258 and 584 leaving remainders 6 and 8 respectively.

Find the GCD using the method of prime factorisation: (A) 115, 207

(B) 159, 1060 (C) 72, 180, 504

(D) 300, 900

(E) 90, 126, 198

Find the GCD of the following numbers using the division method: (A) 900, 550 (B) 1896, 1106 (C) 339, 781, 1004

(A) Find the lowest common multiple of 3, 6 and 7.

3.

(A) Find the rod of the greatest length which can measure 216 m, 72 m and 81 m exactly. What is the number of rods of equal size in each length? (B) Two payments are made of Rs. 276 and Rs. 1242 with cheques of equal amount. What is the largest possible amount of each cheque? How many cheques are paid in all?

(D) 66, 121, 1221 (E) 3120, 1632, 5610 www.betoppers.com

32 4.

Reduce the following fractions to the lowest terms by canceling the H.C.F. of the numerator and the denominator. (A)

(C) 5.

144

(B)

198 2401

(D)

1379

2211

8.

9.

(B) 16

(C) 24

(D) 42

Determine all the prime factors of 138.

5025

(A) 2, 3 and 23

(B) 1, 2 and 23

4130

(C) 2, 3 and 46

(D) 6 and 23

7021

6.

(A) What is the H.C.F of two consecutive numbers?

(A)A rectangular courtyard is 9 m 18 cm long and 8 m 16 cm broad. It is to be paved with square stones of the same size. Find the least possible number of such stones. (B) Two baskets contain 276 and 1242 apples respectively. Find the largest number of apples that can be taken out from each basket every time to make them empty.

7.

(A) 12 5.

(B) Find the least number which when divided by 18, 30, 48 and 54 leaves the remainder 5. 6.

4.

6th Class Mathematics 3 is a common factor of 9 and x. Among the following, which is not a possible value of x?

Four bells toll at intervals of 4, 7, 12 and 84 seconds. The balls toll together at 5 o’clock. When will they again toll together? A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again? A boy saves Rs. 4.65 daily. Find the least number of days in which he will be able to save an exact number of rupees saved.

10. Determine the two numbers nearest to 10000 which are exactly divisible by each of 2, 3, 4, 5, 6 and 7.

7.

The prime factors of 100 are (A) 1, 2 and 5

(B) 2 and 5

(C) 4 and 5

(D) 5 and 10

For the sequence 2, 6, 10, x, 18, the value of x is (A) 12

8.

(C) 15

(D) 16

23, x, 31, 37, 41, y, 47 The list of numbers above shows prime numbers arranged in an increasing order. The value of x + y is (A) 68

9.

(B) 70

(C) 72

(D) 74

Find the lowest common multiple of 6, 14 and 16. (A) 2

(B) 14

(C) 16

(D) 336

10. 7 and 9 are factors of x. What is a possible value for x? (A) 63

(B) 70

(C) 81

(D) 84

11. 67, 61, p, q, r, s, 41 is an arrangement of prime numbers in a decreasing order. Among the following, which is the number 53? (A) p

(B) q

(C) r

(D) s

12. The lowest common multiple of 3, 6 and x is 24. A possible value of x is (A) 3

IIT JEE Worksheet

(B) 14

(B) 4

(C) 6

(D) 8

13. The common multiples of 20 and 30 are

I.

Single Correct Answer Type

(A) 40, 60

(B) 50, 60

1.

Among the following numbers, which are common multiples of 8 and 24?

(C) 60, 120

(D) 100, 120

2.

(A) 32 and 48

(B) 16 and 72

(C) 48 and 72

(D) 72 and 100

The highest common factor of 14 and 49 is (A) 2

3.

(B) 7

(C) 14

(D) 21

Which of the following numbers are prime factors of 150? (A) 2, 3 and 5

(B) 5 and 6

(C) 5 and 10

(D) 2, 3 and 15

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14. The sum of all the prime factors of t is 5. A possible value of t is (A) 36

(B) 30

(C) 26

(D) 20

15. 19, x, 29, 31, y, 41, 43 are a list of prime numbers in ascending order. The value of x + y is (A) 60

(B) 37

(C) 23

(D) 21

Number System-II

33 (C) 36 = 3  2  2  2 ; 72 = 3  2  3  2  2 ;90 = 3 3 5 2

II. Multiple Correct Answer Type 16. Which of the following statements is true?

(D) 36 = 3  2  3  2 ; 72 = 3  2  3  2  2 ;90 = 3 3 5 3

(A) 1 is a prime number (B) 1 is neither a prime nor a composite number (C) A prime number will have the number itself as one of its factors

25. Find HCF of 36, 72, 90 (A) 16

(A) 3

(B) 6

(C) 53

(D) 11

18. The prime number between 40 and 50: (A) 49

(B) 47

(C) 41

(D) 43

19. The common multiples of 7 and 13: (A) 91

(B) 182

(C) 273

(D) 364

20. Which of the following is the number which can divide 60, 90, 120? (A) 30

(B) 60

(C) 90

(D) 15

21. Which is false? (A) (HCF × LCM) of a and b = a  b (B) (HCF + LCM) of a and b = a + b (C) (HCF – LCM) of a and b = a – b (D) HCF  LCM) of a and b = a  b

Based on this information answer the questions given below.

IV. Integer Type 27. Let N be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then find the sum of the digits in N. 28. A number has two digits whose sum is 10. If 18 is added to the number, its digits get interchanged. Find the HCF. of the number and the new obtained number. 29. The number of possible sets of two prime numbers such that their sum or difference will never yield a composite number is __________. 30. If L.C.M. of two numbers is 5 times of one of the numbers and the other number is 25, then H.C.F. of the two numbers is

V. Matrix Matching (A)

(B) (C)

22. The largest number that divides 36 and 90 without a remainder. (A) 15

(B) 16

(C) 17

(D) 18

23. The smallest number which when divided by 36 and 90 leaves a remainder of 8. (A) 158 (B) 168

(C) 188

(D) 17

(A) 3200 (B) 3240 (C) 3420 (D) 3042

III. Paragraph Type There’s a competition held among three hefty boys, Ram, Rahim and Robert in eating biscuits. Ram could complete eating 36 biscuits, Rahim ended up in eating 72 biscuits and Robert stood as a winner eating 90 biscuits.

(C) 15

26. Find the product of LCM and HCF of 36 and 90

(D) A composite number will have only two factors 17. Which of the following is/are a factor of 1113?

(B) 18

(D)

Column - I Given that the lowest common multiple of 15 and 25 is p. The value of p is The common multiple of 12 are The lowest common multiple of 10, 15 and 20 is x. The value of x is The common multiple of 3 are

Column - II

(p) 90 (q) 48

(r) 63 (s) 75 (t) 60

(D) 186

24. Prime factors for the following: 36, 72, 90 (A) 36 = 3  2  3  2 ; 72 = 3  2  3  2  2 ; 90 = 3  3  5  2 (B) 36 = 3  2  3  2 ; 72 = 3  2  2  2  2 ; 90 = 3  3  5  2

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6th Class Mathematics

34

Fractions Example:

1. Understanding a Fraction A fraction is a number representing part of a whole. The whole may be a single object or a group of objects and each to be equal.

5 1 2 , , fractions does not have any 12 2 3 common factors in both numerator and denominator. (i)

p

They are written in lowest terms.

Definition: The numbers of the form q , where ‘p’ and ‘q’ are whole numbers and q  0 are called fractions.

6 10 , have common factors in 8 12 both numerator and denominator

(ii) Fractions

1 2 1 3 8 , , , , etc are fractions, here 2 3 4 5 12

p We know that the numbers of the form q

1 means the whole thing can be divided into 2 two equal parts and we taken out 1 part similarly

where p,q w (whole numbers), q  0 and p,q does not have any common factors are called rational numbers.

Example:

3 means the whole thing can be divided into 3 three equal parts and we taken out two parts.

1 2 4 10 Example: , , , ,etc are 2 3 5 11 numbers.

rational

Therefore ‘every rational number is a fraction’

2. Simplest Form of a Fraction In the above figure the whole rectangle can be divided into 5 equal parts among which 2 parts are shaded. The shaded portion represents twofifths and is denoted by fractional number and

2 . Here two-fifths is a 5 2 is a fraction. 5

If the numerator and denominator of a fraction have no common factor except 1, then the fraction is said to be in its simplest form or in lowest terms. Or if a fraction is said to be in simplest form if the H.C.F of its numerator and denominator is 1.

I.

If a fraction is in simplest form then it is called irreducible fraction.

How to Read Fractions 1 2 1 read as one-half; as two-thirds; as 2 3 4

Irreducible Fraction

II.

Reducible Fraction If a fraction is not irreducible fraction then it is called reducible fraction.

3 8 as three-fifth and read as 5 12 eight-twelths and so an.

III. Method to find Simplest Form of a Fraction

p Note: For a given fractions q , p is called the numerator.Where p,q may have common factors

We can reduce given fraction into its simplest form using any of the following two methods. Method 1:

5 1 2 6 10 Example: , , , , ,etc , 5 is numerator 12 2 3 8 12 and 8 is denominator

Divide numerator and denominator of given fraction by their H.C.F 9 Example: To find simplest form of ; the 15

one-quarter;

Fractions can also be written in its lowest terms.

H.C.F of 9 and 15 is 3. Now divide both 9 numerator and denominator of by 3 to get 15 reduce it into simplest form. Hence the simplest 9 3 form of is . 15

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5

Number System-II

35

Method 2: In this method, we can divide both numerator and denominator of the given fraction by common factor till we are left with common 1 only. 72 Example: We will find simplest form of 90

Example:

1 1 1 1 , , , etc, are all unit fractions. 2 3 4 12

IV. Proper Fractions A fraction in which the numerator is less than its denominator is called proper fraction. Example:

using this method or follows, . Therefore the 72 4 simplest form of is . 90

5

1 2 4 8 , , , etc, are proper fractions 2 3 5 12 lying between 0 and 1 on a number line.

3. Representing Fractions on Number Line 1 on a number line, 2 we can divide the length between 0 and 1 into two equal parts and we can take 1 part as a To represent given fraction

V.

Improper Fractions A fraction in which the numerator is greater than or equal to its denominator. Note: The numerical value of all the improper fraction does not lie between 0 and 1 on a number line.

1 fraction . 2 1

0

VI. Mixed Fraction A combination of a whole number and a proper fraction is called mixed fraction.

1 2

Example:

2 on a number line, we can divide 3 the length between 0 and 1 into three equal parts and we can take into three equal parts and we

1 1 2 2 , 3 , 16 etc, are all mixed fraction. 3 4 3

To represent

can take 2 parts out of 3 as a fraction

we can write mixed fraction = whole number part + fractional parts = whole

2 3

number +

1 In a mixed fraction 2 , 2 is whole number 3

1

0 1 3

2 3

and

2 3

4. Classification of Fractions I.

1 1 1 is a proper fraction. 2  2  3 3 3

VII. Equivalent Fraction

Like Fractions

Two or more fractions representing the same point of a whole are called equivalent fractions.

Fractions with the same denominator are called like fractions. Example:

1 2 3 4 5 6       .....are 2 4 6 8 10 12 equivalent fractions Eg:

1 2 4 8 16 , , , , are like fractions. 9 9 9 9 9

II.

Numerator of fractional part Denominator

Unlike Fractions Fractions with different denominators are called unlike fractions. Example:

1 2 3 5 , , , etc, are unlike fractions. 2 3 5 8

III

1 2

2 4

3 6

Unit Fractions Fractions with 1 as numerator are known as unit fractions. www.betoppers.com

6th Class Mathematics

36 From the figures in last page, the shaded regions of each figure are equal i.e.,

L.C.M of 4, 6, 9, 8, 12 = L.C.M of 8, 9,12 = 72

1 2 3   are equivalent fractions. 2 4 6

Now we are equating denominator of all fractions to 72 by multiplying numerators and denominators of all given fractions with a suitable number.

Finding Equivalent Fractions: To find an equivalent fraction of a given fraction, we can multiply or divide both numerator and denominator of the given fraction by the same non-zero number. Examples:

Therefore

3 3  18 54 5 5  12 60   ;   ; 4 4  18 72 6 6  12 72 4 4  8 32   ; 9 9  8 72

1 1 2 2 2  3 6   ;  2 2  2 4 4  3 12

1 1 9 9 7 7  6 42     and 8 8  9 72 12 12  6 72

1 1 2 2  3 1 3 1 5      ..... are 2 2 2 43 23 25 equivalent fractions 12 12  2 12  3 12  6    18 18  2 18  3 18  6 are also equivalent fractions Note: If two fractions are said to be equivalent fractions, the product of the numerator of the first and denominator of the second is equal to the product of denominator of the first and the numerator of the second. These products are

p r called cross products. If q , s are two fractions then their cross pr oduct is denoted as p q

r s

and if ps = rq then we say that

p r , q s are

equivalent fractions.

1 2 , are equivalent fractions since 2 4 1×4=2×2 4=4 Example:

5. Interconversion of Fractions I.

Conversion of Unlike Fractions into Like Fractions To convert unlike fractions we can make denominators of all given fractions equal to their L.C.M (i.e., we convert each of the given fractions into an equivalent fractions having a denominator equal to the L.C.M of all the denominators of the given fraction.) Example: To convert the unlike fractions 3 5 4 1 7 , , , , into like fractions first we 4 6 9 8 12

find L.C.M of denominator of given unlike fractions. www.betoppers.com

54 60 32 9 42 , , , , are the 72 72 72 72 72 required like fractions. Therefore

II

Conversion of mixed Fraction into Improper Fraction We know that mixed fraction has two parts; one is whole number part and the other is fractional part. To convert mixed fraction into improper fraction, multiply the whole number part with the denominator of the fractional part and add the product to the numerator of the fraction part. This gives the numerator of the improper fraction and its denominator is same as denominator of fraction part. i.e., if mixed fraction = whole number

Numerator , then the improper fraction Denominator can be expressed as Improper Fraction =

 Wholenumber  Denominator  Numerator Denominator

Example: Convert mixed fraction 5

2 into 3

improper fraction as follows: 2 2  5  3  2 15  2 17 5 5   3 3 3 3 3

III. Convert an Improper Fraction into a Mixed Fraction We know that improper fraction has greater numerical value of numerator than denominator. To convert an improper fraction into a mixed fraction, divide the numerator by denominator then the quotient so obtained forms the whole number part and the remainder forms numerator of fractional part.

Number System-II

37

Here denominator of fractional part is same as denominator of given improper fraction. So that mixed fraction can be written as follows

Remainder Mixed Fraction  Quotient Divsior 23 into mixed 7 fraction as follows divide 23 by 7 as follows Procedure to convert

7)23(3 21 2 We have quotient 3 and remainder 2 when 23 divided by 7. Therefore required fraction  Quotient

2 2 2 2 and ;  9 6 6 9 since both have same numerator and 6 28 Now



Example:

2 14 14 2  since For the fractions , ; 9 9 9 9 both have same denominator and 14 > 2 (ii) Fractions with same Numerator Among two fractions with same numerator. The fraction with smaller denominator is greater than the other.

30 28 6 4    35 35 7 5

(B) To compare fractions with different numerators and different denominators we can also use the following method for the fractions

a c , find cross products ad and bc. b d If ad > bc, then then

Fractions with same Denominator Among the fractions with same denominator, the fraction with greater numerator is greater than the other.

4 4  7 28   ; 5 5  7 35

a c  ; If ad < bc, b d

a c a c  ; If ad = bc, then  b d b d

Example:

3 8 and by 5 11 considering cross multiplication Compare the fractions 3 5

8 11

The products are 33, 40 and 40 > 33

 5 × 8 > 3 × 111  Therefore

8 3  11 5

8 3  11 5 www.betoppers.com

6th Class Mathematics

38

7. Arranging fractions in Ascending and Descending Order Ascending order of fractions means arranging fractions from smaller value to greater value and descending order of fractions means arranging fractions from greater value to smaller value. This can be done by comparison of fractions. Examples: Rearrange the following fractions in ascending order and descending order

2 4 7 11 23 , , , , 3 5 15 20 30 Here the given fractions have different numerators and different denominators. L.C.M. of 3, 5, 15, 20, 30 = 2 × 3 × 10 = 60 Therefore

2 2  20 40 4 4  12 48   ;   ; 3 3  20 60 5 5  12 60

7 7  4 28 11 11 3 33   ;   and 15 15  4 60 20 20  3 60

Therefore sum of all like fractions =

Sum of all Numerators of Fractions Common Denominator Example: 5 8 13 22 2  8  13  22 48      8 6 6 6 6 6 6 Addition of Unlike Fractions: Step1: We find L.C.M of denominators of all fractions. Step2: Now convert each of the given fractions into equivalent like fractions by equating their denominators to L.C.M of denominators. Step3: Now we can add all like fractions which are so obtained in Step2 Step4: Reduce the fraction obtained in Step3 into its lowest terms and convert it into mixed fractions if it is a improper fraction. Examples: Step1: Find the sum of

23 23  2 46   . 30 30  2 60 Since 28 < 33 < 40 < 46 < 48

Step2: Now

5 8 15 , , 9 12 16

5 5  16 80   ; 9 9  16 144



28 33 40 46 48     60 60 60 60 60

8 8  12 96 15 15  9 135   ;   12 12  12 144 16 16  9 144

7 11 2 23 4     15 20 3 30 5

Step3: Now sum =



Therefore the ascending order of given fractions is

7 11 2 23 4 , , , , and the descending order 15 20 3 30 5

4 23 2 11 7 of given fraction is , , , , . 5 30 3 20 15

5 8 15 80  96  135 311     is a 9 12 16 144 144 improper fraction Step4: Therefore

5 8 15 311 23    2 9 12 16 144 144 Addition of Mixed Fractions:

8. Fundamental Operations

 +, -, ×, ÷  on Fractions (i)

Addition Addition of two or more fractions is possible, when they are like fractions. Addition of Like Fractions: If the given set of fractions have same denominator (i.e., like fraction), then the numerator of sum of all fractions is the sum of numerators of given fractions and denominator is their common denominator.

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If given set of fr actions are mixed fractions; Step1: Convert each of the mixed fractions into an improper fraction. Step2: Add all improper fractions using the procedure, either like fractions addition or unlike fractions addition. Example:

2 3 4 Find the sum of 1  2  3 5 5 5

Number System-II

39

II.

Method 1

Subtraction

In above the addends are mixed fractions

Subtraction of Like Fractions

2 7 3 13 1  ; 2  ; 5 5 5 5

Difference of like fractions is a fraction which is having numerator as difference of numerator and denominators is common denominator.

4 19 3  5 5

Now,

2 3 4 7 13 19 39 4 1  2 3     7 5 5 5 5 5 5 5 5 Method 2

 2 3 4  2 3 4 1  2  3  1  2  3    5 5 5 5   5 4 6  5 5  6 1 

7

4 5

Therefore difference of like fraction



Difference of Numerators Common Denominator

Examples:

9 4 94 5    13 13 13 13 2 1  3 5 Subtraction of Unlike Fractions First we find L.C.M of denominators of given fractions.

4 5

Example:

2 3 1 Find sum of 1  2  3 3 4 5

Convert each of the given fraction into equivalent like fraction by equating their denominators to L.C.M of denominators. Now we can find difference between like fractions so obtained.

Method 1 In above addends are mixed fractions

2 5 1  ; 3 3

2 1  3 5

Method 1

3 11 1 16 2  ; 3  4 4 5 5

L.C.M of 3,5 = 15

2 2  5 10   3 3  5 15

2 3 1 5 11 16 Now, 1  2  3    3 4 5 3 4 5 

1 1 3 3   5 5  3 15

 5  20   11 15   16  12  60

100  165  192 457 37   7 60 60 60 Method 2

2 3 1  2 3 1 1  2  3  1  2  3      3 4 5  3 4 5 6

6

 2  20    3  15   1  12  60

 40  45  12 

 6 1

60

37 37 7 60 60

6

97 60

2 1 10 3 10  3 7      3 5 15 15 15 15 Method 2 2 1  2  5  1 3  10  3 7   :   3 5 15 15 15

Subtraction of Mixed Fractions While subtracting two or more mixed fractions, convert each of mixed fractions in to an improper fraction and then find the difference of fractions so obtained using any of the above methods. Examples:

1 1 1 1 Find 3  2  4  2 6 4 3 5 www.betoppers.com

6th Class Mathematics Reciprocal or Multiplicative Inverse of a Fraction

40

1 19 1 13 Since 3  ; 4  6 6 3 3

If the product of any two fractions is 1, then each one of them is called reciprocal or multiplicative inverse of other.

1 9 1 11 2  ;2  4 4 5 5 Now

p q p q For the fraction q , p , q  p  1

1 1 1 1 19 9 13 11 3 2  4 2     6 4 3 5 6 4 3 5 (Therefore L.C.M of 6, 4, 3, 5 = 60)

p q Therefore reciprocal of q is p and

190  135  260  132 60 (Since 60 6 = 10; 60 = 15; 60 5 = 2)

q p reciprocal of p is q





3 = 20; 60

4

450  267 186  60 60

3

3 1 or 3 60 20

5 6 is 6 5

The reciprocal of

The reciprocal of 1

1 1 1 1 3 Therefore 3  2  4  2  3 or 6 4 3 5 60 3

Examples:

1 20

2 is 3

2 5 3   Since 1   3 3 5  The reciprocal of 2 is

1 2

III. Multiplication Multiplication of two fractions is a fraction whose numer ator is products of numer ators of given fractions and denominators is product of denominators of given fractions, we can define it as follows:

25    Therefore 2   1  

IV. Division To divide one fraction by another fraction, we multiply the dividend fraction by the reciprocal of the divisor. Examples:

Product of Fractions



Product of their Numerators Product of their Denominator

i.e., for any two fractions

a c a c ac ac , ,   or b d bz d b  d bd

3 2 3 3 9     5 3 5 2 10 2 2 5 12 5 5 25 1 2      3 5 3 5 3 12 36 Note:

Example: For any fraction Multiplication of

2 3 and is 5 8

2 3 6 3    5 8 40 20 For the multiplication of mixed fractions convert them into improper fractions and then consider multiplication. Example:

3 5 11 11 121 2 1    4 6 4 6 24 www.betoppers.com

1

a a a a ,1    1  and b b b b

a b a a a a  ; 1  ;  0  0   0 b a b b b b

Every non zero fraction or rational number has multiplicative inverse. Zero does not have multiplicative inverse.

Number System-II

41

Formative Worksheet 1.

2.

14. Anandini complete

How long would she take to complete the whole project?

6 a 36 = = 11 33 b Determine the values of a and b. Are the following fractions equivalent? (A)

1 1 of her project in 3 days. 6 2

1 5 , 5 25

(B)

3 10 , 8 16

1  14  2 3   . 5  15  3

15. Solve  3

16. There are 1,500 Americans working in a factory. The number of Chinese workers is

3 4 or 5 7

3.

Which fraction is smaller?

4.

1 8 5 Arrange , , in ascending order.. 2 9 6

5.

number of American workers and the number of Indian workers is

3 5 11 9 7 5 101 22 4 9 , , , , , , , , , 7 9 2 16 3 5 10 23 3 4

6.

(A)

19 5

(B)

7.

15 5  Calculate 25 25

8.

1 2 Solve 5  4 3 5

9.

Calculate 3 +

10. Solve 13

30 6

Conceptive Worksheet 1.

2.

(i)

1 4

(v)

15 7

4.

2 litres of oil. It is then 5

(ii)

5 6

(iii)

9 8

(iv)

7 3

Write the prime factorization of each number. (ii) 48

(iii) 80

Write each fraction in lowest terms. (i)

7 1 5 . 8 3

11. An oil tank contains 30

Graph the fraction on a number line.

(i) 12 3.

2 1 +7– . 3 4

5 the number of Chinese 9

workers. How many workers are working in the factory?

From these fractions:

Select the (A) proper fractions (B) improper fractions. Convert the following improper fractions into mixed fractions.

3 12

(v)

64 80

(ix)

108 120

(ii)

5 20

(vi)

88 51 225 (vii) (viii) 132 170 375

(iii)

18 4

(iv)

25 55

Multiply/Divide, write the product in lowest terms. (i)

7 2 . 8 3

(ii)

2 5  3 6

of oil is pumped out from the tank. What is the volume of oil left in the tank now, in litres?

(iii)

6 12  15 5

(iv)

10 4  11 5

11 12. A factory has 150 workers. of them are 15

(v)

3 5 × 8 12

(vi)

25 18 30   36 10 50

filled with 37

2 1 litres of oil. Later, 56 litres 3 9

male. How many female workers are there in the factory? 13. Solve

3 7 49   . 4 18 27

3 the 5

(vii)

2 6 16   3 4 15 4 5

(ix) 2  2

1 7

(viii)

64 50  25 8

(x) 3

13 5  6 5 20 10

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6th Class Mathematics

42 5.

Simplify: (i)

3 2  7 7

(ii) 

10 4  (iii) 11 11

1 7 (iv)  8 8

3 4  7 6

(v) (vii)

(xi)

1 6

(viii) 4  3

1 1 1 8 7 2 3 7

(x)

5 4  8 7 4 5

(xiii) 13  2

11 5 2   (xv) 10 7 5

5 12

7 1  10 5

(xiv) 4

5 2 2 6 3

7 1 1 (xvi)   8 3 4

1 5

(xix) 11 6.

(viii) 4 

(ix)

34 5 7

(x) 6

7 21  11

(vi)

7.

(iv) 8.

5 7

(ii)

7 7 , 12 15

7 9

(v)

30 45

(iii)

1 2

25



32 100

14

6





18

18 54   66 132 

36  56 168

12 30 1 20

(ii) 3

7 9

(v) 8

1 3

(iii) 11

310 25

(ii)

37 11

25 3

(v)

235 10

(iii)

56 19

12 15

1 1 2 3 , , , 6 15 5 4

(i)

(iii)

11 4 1 23 , , , 28 7 2 56

(v)

1 4 5 1 , , , 6 5 6 2

(ii)

2 1 5 3 , , , 3 6 12 4

(iv)

3 4 1 5 , , , 5 7 3 12

13. Arrange in descending order:

7  26 (ii) 13

(iii)

3 4  7 9

(v)

7 5 times of (vi) 5  25 20 14

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(iv)

12. Arrange in ascending order:

Evaluate:

3 (i) of 16 5

45 51

1  2 20

10. Convert into improper fractions:

(i)

Write 5 equivalent fractions for the given fractions: (i)



11

(vii)

(iv) (ii)



(ii)

11. Convert into mixed numbers:

1 1 1 7 6 10 20 10

1 5 , 4 6

5 7 1 12 36

36  60 5

(v)

Find LCM of the fractions below: (i)

15

(iii)

(iv) 21

1 1 1 2 4

1 8

Fill in the blanks:

(i) 6

1 1 1 (xvii) 8  9  7 9 8 6 (xviii) 17  8

4  16 7

(i)

5 2  8 9

(xii)

1 3

9.

1 7  5 10

(vi)

4 3 2   5 10 7

(ix) 18

1 1  2 2

(vii)

(iv)

3 11 of 4 17

5 3 1 1 , , , 16 8 4 8

(ii)

1 2 1 1 , , , 4 5 3 9

(iii)

1 1 1 1 , , , 2 3 4 5

(iv)

4 2 5 3 , , , 5 3 8 10

(v)

23 7 47 3 , , , 25 10 50 5

(i)

1 11

Number System-II

43 2.

Summative Worksheet 1.

1

2 25

The diagram below shows a circle which is divided into 16 equal parts. Unshade

What is the mixed number represented by A on the above number line?

5 of the 8

(A) 2

whole circle.

3.

3.

4.

5.

6. 7.

8.

9.

There are 24 hours in a day. What fraction of a day does 11 hours represent? In a family with 11 children, there are 4 boys and 7 girls. What fraction of the children are girls? A work shift for an employee at Mc Donald’s consists of 8 hours. What fraction of the employee’s work shift is represented by 6 hours? There are 35 students in a biology class. If 10 students made grade A in the first test, what fraction of the students made grade A? Four out of 10 marbles are red. What fraction of marbles are not red? There are 100 centimeters in 1 meter. What fraction of a meter is 20 centimeters?

4.

4 2 1 3 (B) 2 (C) 2 (D) 2 5 5 2 4

1 3 5   = 3 4 6 (A) 1

2.

3

A

5 7 9 11 (B) 1 (C) 1 (D) 1 12 12 12 12 1 of his assessment on Monday 5

Asri completed and

3 of it on Tuesday. How much of his 4

assessment did Asri complete in the two days? (A) 1

5.

(B)

A rope, 36

4 9

(C)

3 20

(D)

19 20

1 m long, was cut into three parts 3

measuring 12

2 1 4 m, 13 m and 5 m 5 2 15

respectively. What was the length of rope left?

2 7 3    7 2 4

(A) 5

1 5 1     2  6 12 

10. An estimate for an adult’s waist measurement is found by dividing the neck size (in inches) by

6.

1 . Revanth’s neck measures 18 inches. 2 Estimate his waist measurement.

3

1 5 × = 5 8

(A)

7.

1 2 1 2 (B) 7 (C) 9 (D) 11 6 5 6 5

1 8

(B)

(C)

1 2

Anu had Rs. 150. She gave

HOTS Worksheet Rajani gave

1.

1 5

(D) 2

3 of this to Rajani. 5

1 of the money she received to 2

Narmada. How much did Narmada receive if she also received Determine the fraction of the shaded parts in the figure shown. (A)

2 5

(B)

11 25

(C)

12 25

(D)

4 5

had left? (A) Rs. 90 (C) Rs. 75

1 of the money that Anu 2 (B) Rs. 80 (D) Rs. 45

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6th Class Mathematics

44 8.

9.

2 2    =  7 21 

30 – 

5.

(A) 312

(B) 29

(C) 28

(D) 27

143 147

(A)

6.

 3 5 1 =     4 8  16 (A) 22

10. 7

(B) 20

(C)

1 20

(D)

2 34 1 (B) 9 (C) 8 9 81 8

1 22

(D) 5

1 8

7.

1 1  1 11.     =  5 10  18 (A) 9

(B) 7

(C)

1 5

(D)

1 9

8.

IIT JEE Worksheet Single Correct Answer Type

1.

2 3 5 10     = 7 4 8 1 1 1 1 (A) 10 (B) 12 (C) 14 (D) 16 7 7 7 7 In a class of 50 students,

2 travel to school by 5

bus, 10 travel by car and the rest walk. What is the fraction of students who walk to school?

1 (A) 5 3.

2 (B) 5

4 (D) 5

4 1 1  3 = 9 9 3 2 2 (A) 3 (B) 2 3 3

4.

3 (C) 5

2 (D) 3

In a class of 50 students, 15 are girls. 5 of the

2 of the boys were chosen to 7 represent their class in a game. The total number of students chosen is girls and

(A) 12 (B) 15 www.betoppers.com

(C) 19

(C)

2 7

(D)

Ahmed gave Muthu Rs. 450. If

(A) Rs. 250

(B) Rs. 240

(C) Rs. 230

(D) Rs. 220

6 7

1 2 and of 9 5

2 5 1 3 1  = 5 6 5 (A) 5

3 19

(B) 5

5 19

(C) 5

1 30

(D) 6

1 30

A class has 40 students. Every student

(D) 25

3 of 10

the total fund was used by the class to buy prizes for a Mathematics quiz. How much money does the class fund have left?

9.

(A) Rs. 30

(B) Rs. 50

(C) Rs. 70

(D) Rs. 90

5 2 1   = 18  3 6  (A)

4 9

(B)

5 9

(C)

7 9

(D)

8 9

2  12  1  3  = 3  69  2

10.  7

(A)

2 (C) 1 3

1 5

(B)

contributed Rs. 2.50 to the class fund.

I.

2.

1 4

the money was spent on clothes and food respectively, how much money does Muthu have left?

2 1 5    = 9 3 9

(A) 22

3 5 1    = 8 3 6

1 3

(B)

2 3

(C) 1

1 3

(D) 2

2 3

II. Multiple Correct Answer Type 11. The fraction equal to

(A)

3 14

(B)

6 14

3 is : 7

(C)

9 21

(D)

12 28

Number System-II 12.

1 2  because : 6 3 (A) (1 × 3) < (2 × 6) (B) We cannot say without number line (C)

2 4  3 6

17.

(D) numerator 1 < numerator 2 13.

18.

4 3  , then 7 4 (A)

4 43 3   7 74 4

(B)

4 7 3   7 11 4

4 43 3 4 43 3    (D)  7 74 4 7 7 4 4 14. Which of the following statements is correct. (C)

(A)

ab + c = a +c b

a +c b (B) a +c = c

ab +a c b + c = ad d

(D)

19.

20.

45 In one year, the Rudra’s family drove 12,000 km in the family car. Based on this information how many of these km might we expect to fall in each category below? Work (A) 3840 km (B) 3048 km (C) 3480 km (D) 3804 km Shopping (A) 1044 km (B) 1440 km (C) 1220 km (D) 1022 km Family business (A) 2400 km (B) 2044 km (C) 1044 3480 km (D) 1400 km Medical (A) 110 km (B) 101 km (C) 120 km (D) 102 km

IV. Integer Type

a(b + c) b + c = a d d 15. Which of the following statements is true?

21.

1 2 + = 3 3

(A) 5 hours is

1 of a day 6

22.

9 6  = 3 3

(B) 4 days is

4 of a week 7

23.

3 8 × = 4 3

24.

5 1 ÷ = 4 4

(C)

(C) 1 month is

1 of a year 12

(D) 6 months is

1 of a year 2

V. Matrix Matching

2 16. 3 is equal to: 5

Column I

(A) 3 

2 5

(B) 3 

(C) 3 

2 5

(D)

2 5

17 5

III. Paragraph Type

Work

Vacation other Shopping

3 50

3 25

Social/ recreational

(B) Write 25

(D)

2 as in improper fraction? 5

138 as a mixed number. 5

3 3 3 7 4 4  4 4 5

(p)

127 5

(q)

27

(r)

63

(s)

299 500

(t)

7

3 5

18 20

1 Medical 100

8 25

3 25 13 100

Which of the following fractions is (A) nearest to 3 ? 5

(C) Write

The following chart shows the fractional part of a car’s total mileage in each category of Rudra’s family.

Column II

1 5

Visit friends

2 School/church 50 Family business

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6th Class Mathematics

46

Decimals 1. Decimals and decimal fractions Representing fractions of

1 of 10 parts =

1 1 and as 10 100



decimals and vice versa

1 = 0.1 10

8 = 0.1 × 8 = 0.8 10

Hence, decimals interchangeable.

Decimals are fractions whose denominator is a multiple of 10, that is 10, 100, 1,000, .... and so on.

and

fractions

are

Representing fractions with denominators 10, 100 and 1,000 as decimals

In the figure below, the shaded areas represent

Any fractions with denominators 10, 100 and 1,000 can be expressed in decimals.

8 8 of 10 parts, that is parts. 10

81 = 0.81 100 2

73 = 2.073 1, 000

How to read and write decimals to thousandths?

1 1 1 1 1 10 10 10 10 10 0.1

0.1

4 = 0.4 ; 10

1 1 1 1 1 10 10 10 10 10

3 = 0.03 100

1,987 = 1.987 1, 000

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

2. Place Value Chart of Decimals The place-value chart of decimals as shown below Ten Thousands Thousands 10000

Hundreds

Tens

Ones

100

10

1

1000

Decimal Point

Tenths

Hundredths Thousandths

1 10 (0.1)

1 100 (0.01)

1 1000 (0.001)

We arrange a decimal number 3526.812 in the place value chart as follows.

Thousands

Hundreds

Tens

Ones

1000

100

10

1

3

5

2

6

Decimal Point

Tenths

Hundredths

Thousandths

1 10 8

1 100 1

1 1000 2

Therefore place value of 3 = 3000 place value of 5 = 500, 2 = 20, 6 = 6 place value of 8 =

8 = 8 tenths 10

1 place value of 1 = = 1 hundredths 100

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place value of 2 =

2 = 2 thousandths 1000

Therefore the expanded form of 3526.812 is 3526.812 = 3000 + 500 + 20 + 6 +

8 1 2   . 10 100 1000 Here 3526.812 is called ordinary or short form.

Number System-II

47 Example:

3. Decimal Places The number of digits in the decimal part of a decimal gives the number of its decimal places. Example: 4.25 has two decimal places 6.125 has three decimal places 4.6 has only one decimal place 4.0 has no decimal place

4.5, 4.25, 1.52512, 32.3335 etc are terminating decimals. Non-Terminating Decimals If one or more digits in a decimal part of a decimal are repeated again and again without any end, they are called non-terminating decimals.

Classification of Decimals according to its Decimal Places

Example:

There are two types of decimals according to its decimal places, they are

(i)

1  0.333... 3

(ii)

10  0.91909090... 11

(i) Like decimals

(ii) Unlike decimals

Like Decimals Decimals having the same number of decimal places are called like decimals.

(iii) 1.732732732...etc are non-terminating decimals

Example: 4.25, 6.75, 10.02 are like decimals.

The digit/s which are repeated again and again without any end in a decimal is called the ‘limit’ of the decimal.

Unlike Decimals Decimals having different number of decimal places are called unlike decimals. Example: 4.2, 4.25, 5.625 etc are unlike decimals

In above example (i)

The limit of 1.732732..... is 732

(ii) The limit of 0.333...is 3

Important Result:

(iii) The limit of 0.9090.... is 90

For decimal 0.2, 0.20, 0.200

While representing non-terminating decimals we can write a line on the digits which are repeated continuously. From above examples

Case(i)

0.2 

2 2  10 20   and 10 10  10 100

20 0.20  . therefore 0.2 = 0.2z0 100

(i)

1  0.333... can be written as 0.3 3

Case(ii) Now 0.20 

20 20  10 200   100 100  10 1000

(ii)

10  0.9090 can be written as 0.90 11

and 0.200 

200 therefore 0.20 = 1000

(iii) 1.73273..... can be written as 1.732

0.200 From case (i) & (ii) we have 0.2 = 0.20 = 0.200 = ... etc Therefore on putting any number of zero’s to the extreme right side of the decimal part of a decimal does not change the value of the decimal. We can write 4.12 = 4.120 = 4.1200 etc. In this way we may convert unlike decimals into like decimals. Terminating Decimals

or

Non-Terminating

Terminating Decimals: If the digits coming after decimal point are finite, then it is called terminating decimal with terminal place. These decimals are called terminating decimals or decimals with an end.

4. Inter- conversion of Decimal and Fractions I.

Decimal to Fraction Write decimal fractions without decimal point as the numerator of the fraction Now write 1 followed by as many zeros as there are as many digits after decimal point as the denominator of the fraction. Now write the simplest form of fraction so obtained Examples: i)

25 1  , Hence after decimal 100 4 point there are two digits. 0.25 

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6th Class Mathematics

48 ii)

1.23 

123 100

Example: Convert

25732 6433  There are 1000 250 three digits after decimal place.

Dividend

(iii) 25.732 

II.

Fraction into a Decimal

41) 10 ( Divisor

Step1:

11) 100 (0.9 99 1  Remainder

Step2:

11) 100 (0.9 99 10

Step3:

11) 100 (0.9 99 100 99 1  Re mainder

We convert a general fractions into a decimal by division method as follows: When a given fraction is a proper fraction Example: Find

1 into decimal 4

Given fraction is proper fraction, denominators does not divides numerator

into decimal

Step1: Plot decimal point in quotient and put a zero on a right of the dividend then divide it just as whole number.

Here 11 does not divides 10 Dividend 4) 1 (  Quotient

Step4: If the same remainder coming repeatedly then terminate the process and write quotient as required decimal.

Divisor

Dividend

Quotient 4) 10 ( 0.2 8 2  Remainder

11) 100 (0.9 99 100 99 100 99 1  Re mainder

Step2: If we get non-zero remainder, then

put a zero on a right of the remainder and then check whether it is divided by divisor or not, if it is divisible then divide by actual division. 4) 10 (0.25 8 20 20 0  Re mainder

In above, the remainder 1 repeats continuously so that the division terminates at same end. Therefore the required decimal is non-terminating decimal . Therefore

10  0.9090.....  1.90 11

III. Improper or Mixed Fraction into a Decimal

Continue this process until getting remainder zero and quotient is required decimal for given fraction.

If a given fraction is mixed, then convert it into proper fraction. Then follow the following steps.

Step 3: If the number so obtained in (Step2) above step is not divided by divisor, then put another zero as a right of the remainder, as well as put a zero on right of the quotient also, then check whether it is divisible by divisor. It is divisible then follow actual divisor otherwise continue Step3. Continue this process until getting remainder zero. Now the quotient is required decimal the following examples explains Step3 www.betoppers.com

Step1: Divide the dividend by divisor using actual division, then we get quotient as well as non-zero remainder which is not divided by divisor. Now we follow the steps which are given for conversion of proper fractions into decimal. Examples: Convert 9

7 into decimal 8

Number System-II

49 Compare 32.416 and 32.418

7 79 Given mixed fraction  9  8 8

Here the whole number parts and the digits at tenths, hundredthsplaces are equal, now compare digits at thousandths place clearly 8>6

By actual division, we have Step1: Divisor  8) 79 (9  Quotient 72 7  Re mainder

Step2: 8 ) 79 ( 9. 79 70

8 ) 79 ( 9. 8 72 70  64 6  Remainder

8 ) 79 ( 9. 8 72 70 64 60

8 ) 79 ( 9. 87 72 70 64 60 56 40



8 ) 79 ( 9. 87 72 70 64  60 56 4  Remainder 8 ) 79 ( 9. 875 72 70 64 60  56 40 40 0

5. Comparison of Decimals To compare two or more decimals, we take the following steps. Step1: First we convert the given decimals into like decimals Step2: Now compare the whole number parts, the decimal with the greater whole number part is greater Step3: If the whole parts are equal, compare the tenths digits. The decimal with the greater digit in the tenths place is greater. Step4: If the tenths digits are also equal, compare the hundredths digits, continue this process until getting unequal digits. Examples: Compare 76.82 and 58.70 First compare whole number parts clearly 76 > 58 Compare 2.78 and 2.52, here whole number parts are equal now compare the digits at tenths place. Clearly 7 > 5. Therefore 2.78 > 2.52

Therefore 32.418 > 32.416

6. Ascending or Descending Order of Decimals The arrangement of decimals from smaller decimal to greater decimal is called ascending order of decimals. It can be done by comparison of decimals. The arrangement of decimals from greater decimal to smaller decimal is called descending order of decimals. It can be done by comparison of decimals. Examples: Arrange 5.37, 3.57, 3.5, 5.3, 5.7, 3.07 in ascending and descending order. Given decimals can be written as like decimals 5.37, 3.57, 3.50, 5.30, 5.70, 3.07; first we compare the decimals whose whole number parts are equal. Clearly 3.07 < 3.50 < 3.57 and 5.30 < 5.37 < 5.70 Now compare 3.57 and 5.30 clearly 3 < 5 Therefore 3.57 < 5.30 Hence 3.07 < 3.50 < 3.57 < 5.30 < 5.37 < 5.70 Therefore the ascending order is 3.07, 3.50, 3.57, 5.30, 5.37, 5.70 and the descending order is 5.70, 5.37, 5.30, 3.57, 3.50, 3.07.

Rounding off Decimals The decimal place (d.p.) of a decimal is the number of digits to the right of the decimal point. A decimal can be rounded off, correct to a certain number of decimal places (d.p.). The method of rounding off a decimal is similar to the method of rounding off a whole number. To round off a decimal to a specific number of decimal places: Look at the digit on the right of the specific digit to be rounded.

If the digit on the right is  5.

Add 1 to the digit to be rounded off and omit all other digits on its right.

If the digit on the right is < 5.

Maintain the number to be rounded off and omit all the other digits on its right.

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6th Class Mathematics

50

7. Four Fundamental Operations

 +, -, ×, ÷  on Decimals I.

Therefore 5.36 x 12 = 64.32

Addition

Multiplication by 10, 100 and 1000

First convert the given decimals into like fractions.

When a decimal number is multiplied by 10, then the decimal point moves to the right by one place.

Now write the addends one under the other in column form so that the decimal points of all addends are in the same column and the digits at the same place are in the same column Now add decimals as actual addition and then put decimal point directly under the decimal points in the addends Examples: Add 5.6, 3.59, 12.56, 123.56. First convert all decimals into like decimals 5.600, 3.590, 12.056, 123.560 Therefore the sum of given decimals is 144.806

II.

The given decimal number has 2 decimal places

Subtraction

Example: 23.56 x 10 = 235.6 When a decimal number is multiplied by 100, the decimal point moves to the right by two places. Example: 23.562 x 100 = 2356.2 When a decimal number is multiplied by 1000, the decimal point moves to the right by three places. Example: 23.562 x 1000 = 23562 Similar way, if a decimal number is multiplied by 10 or power of 10 then the decimal moves to the right by the number equal to the index of 10.

To subtract one decimal from other, first convert the given decimals into like decimals.

Multiplication of two Decimals

Write smaller decimals under greater decimal in column form, so that decimal points of the both the decimals are in the same column. Now subtract decimals by using actual subtract and then put decimal point directly under the decimal points of the given numbers.

Now count the total number of decimal places in both the decimals.

Example: Subtract 12.56 from 25.3 First convert given decimals into like decimals 12.56, 25.30 are like decimals 25.3 – 12.56 =12.74

First multiply the decimal numbers without taking decimal point.

Now in the product obtained when the two decimals multiplied, place the decimal point after leaving the number of digits from right most places to left count is equal to total number of decimal places. Examples: Multiply 12.26 and 5.36 Multiply 1226 and 536 1226 536 657136

Subtract 26.568 from 128.70 26.568, 128.700 are like decimals 128.7 – 26.568 = 102.132

III. Multiplication Multiplication by a Whole Number Multiply the decimal without the decimal point by the whole number. Place the decimal point so as to obtain as many decimal places in the product as there are in the decimal number. Examples: Multiply 5.36 by 12

Sum of decimals places in given two decimals is = 2 + 2 = 4 So, we put decimal point in the product after leaving 4 digits from right most place to left. Therefore 12.26

IV. Division Division by 10,100,1000 When a decimal is divided by 10, the decimal point moves to the left by one place.

First we multiply 536 by 12 Example: www.betoppers.com

5.36 = 65.7136

2.32  .232  0.232 10

Number System-II

51

Similar way, when a decimal is divided by 100, the decimal point moves to the left by two places and moves to the left by three places when divided by 1000. Similarly when a decimal is divided by 10 or power of 10, the decimal point moves to the left by number of digits equal to index of 10. Division of a Decimal by a Decimal Convert the divisor into a whole number by multiplying the dividend and the divisor by a suitable power of 10. Eg:

2.4318 2.4318 100 243.18    0.63 0.63 100 63

9.

10. Solve 49.81 – 10.19 – 3.54. 11. After winning Rs. 2 150 in a Drawing competition, Haneef bought his father a T-shirt for Rs. 75.89 and his mother a batik sarong for Rs. 186.99. How much money does Haneef have left? 12. Find the product of the following. (A) 18.13 × 5

Therefore

(B) 4.56 × 5.6

13. Calculate the following.

Now divide the new dividend by the whole number so obtained. 63 ) 243.18 ( 3. 86 189 841 504 378 378 0

In a grocery store, Bharati bought a tin of biscuits for Rs. 15.95, a bag of rice for Rs. 22.29 and a box of sweets for Rs. 1.65. How much did Bharati pay altogether?

(A) 0.054 × 10

(B) 1.796 × 100

(C) 48.38 × 1,000 14. Find the product of the following. (A) 9 × 0.1

(B) 7.5 × 0.01

(C) 89.4 × 0.001

2.4318 243.18   3.86 0.63 63

15. A table weighs 5.67 kg. What is the total mass of 6 identical tables? 16. Evaluate 75.38  5. 17. Divide the following.

Formative Worksheet 1.

Express the following in decimal form.

7 (A) 10 2.

3.

1 8

(B)

15 8

(A)

5.

The figure below is a number line.

1.

3.75

4

7 100

(B)

23 1000

(C)

3 4

p

Find the value of p. 6.

6 7

Write each of the following fractions as a decimals. (A)

3.25

(B) 0.18 

Conceptive Worksheet

(A) 0.02 (B) 0.075 (C) 1.228 Which of the decimals is greater, 485.760 or 485.670?

5.6 0.008

19. Aunt A, Aunt B and Uncle C divided 11.775 kg of wheat flour equally among themselves. Aunt A used all her wheat flour equally to bake 5 chocolate cakes. How much wheat flour did each cake need?

Convert the following decimals into fractions.

4.

(B) 22.128  0.001

18. Divide the following.

65 34 (B) (C) 5 1, 000 100

Change the following fractions into decimals. (A)

(A) 79.88  100

2.

Round off 87.4592 to (A) the nearest whole number, (B) 1 decimal place,

Write each of the following decimals as a fraction. (A) 0.6

3.

(B) 0.13

(C) 0.425

State the decimal represented by the shaded parts in the following figures.

(C) 2 decimal places, (D) 3 decimal places. 7.

Round off 13.539 to the nearest tenth.

8.

Solve 3.5 + 7.029 + 18.953.

(A) www.betoppers.com

6th Class Mathematics

52 2.

Express 0.07 as a fraction. (A)

7 10

(B)

1 70

(C)

7 100

(D)

1 700

(B)

4.

Change the following fractions to decimals. (A)

1 5

(D) 9 5.

(B)

1 4

(E) 14

(C) 1

2 5

1 8

Complete the given number line 8.16

6.

82 100

8.18

3.

8.19

4. 8.22

Round off each of the following decimals to the number of decimal places given in brackets. (A) 2.1425 (3 d.p.) (B) 0.01721 (2 d.p.)

0.0999 as a fraction is (A)

999 10

(B)

999 100

(C)

999 1, 000

(D)

999 10, 000

The place value of the digit 4 in 0.01541 is (A) tenths

(B) hundredths

(C) thousandths

(D) ten thousandths

P

0

(A) 0.42, 0.5, 0.39, 0.22

(A) P 6.

Arrange the following decimals in descending order.

7.

(A) 3.12, 3.21, 3.1, 3.09 (B) 0.42, 1.01, 0.92, 0.63 (C) 0.99, 10.1, 3.92, 0.097 Round off each of the following decimals correct to the number of decimal places given in the brackets.

8.

9.

0.8

(B) Q

(C) R y

On the above number line, y represents (A) 0.83 (B) 0.85 (C) 0.86 (D) 0.88 Which of the following is the smallest decimal? (A) 0.018 (B) 0.07 (C) 0.074 (D) 0.0054 Which of the following is the greatest decimal? (A) 0.0019 (B) 0.009 (C) 0.019 (D) 0.0091 Which of the following is the nearest to 12.531?

(ii) 3.18 (1 d.p.)

(A) 12.54

(B) 12.53

(iii) 9.186 (2 d.p.)

(iv) 0.9192 (3 d.p.)

(C) 12.52

(D) 12.50

(v) 6.1995 (3 d.p.)

(vi) 0.1495 (3 d.p.)

(vii) 14.178 (2 d.p.)

(viii) 4.096 (1 d.p.)

Summative Worksheet 18 Express 100 000 as a decimal. (A) 0.0000018

(B) 0.00018

(C) 0.018

(D) 0.18

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(D) S 0.9

(i) 0.4192 (2 d.p.)

(ix) 15.972 (1 d.p.)

1.

0.1

On the above number line, which of the letters represents 0.075?

(C) 5.34, 5.43, 7.02, 5.099

9.

S

Arrange the following decimals in ascending order.

(B) 0.042, 0.9, 0.03, 0.0099

8.

R

5.

(C) 52.167 (1 d.p.) (D) 1.0478 (2 d.p.) 7.

Q

10. 1.84359 rounded off to 3 decimal places becomes (A) 1.843

(B) 1.844

(C) 1.845

(D) 1.846

11. Which of the following fractions has the same value as 0.009? (A)

9 100

(B)

1 900

(C)

9 1000

(D)

1 9000

Number System-II

53

12. The diagram below shows a list of decimals. I. 53.760 II. 4.016 III. 2.63 IV. 0.06 Which are the highest and lowest decimals? (A) I and II

(B) II and III

(C) III and IV

(D) I and IV

9.

HOTS Worksheet 1.

2.

3.

4.

5.

6.

7.

8.

Between which two whole numbers on the number line is the number 0.5 lies? (A) 0 and 1

(B) 1 and 2

(C) 2 and 3

(D) – 1 and 0

Between which two whole numbers on the number line is the number 3.3 lies? (A) 0 and 1

(B) 1 and 2

(C) 2 and 3

(D) 3 and 4

1 25

(B)

1 50

(C)

1 100

(D)

1 10

36 25

(B)

72 25

(C)

36 50

(D)

72 100

(C)

1 200

(D)

1 2000

55 m = (A) 0.055 km

(B) 0.55 km

(C) 0.0055 km

(D) 5.5 km

(A) 0.005 kg

(B) 0.05 kg

(C) 0.5 kg

(D) none of these

(A) 5.005 kg

(B) 5.05 kg

(C) 5.5 kg

(D) 0.55 kg

(A) 12.02 kg

(B) 12.2 kg

(C) 12.002 kg

(D) 12.002 kg

IIT JEE Worksheet I.

Single Correct Answer Type

1.

22  10 (A) 0.22 (C) 2.02

2.

1

(B) 111.1

(C) 111.001

(D) 111.0001

2 3 4    10 100 1000

3.

(A) 0.234

(B) 2.34

(C) 23.4

(D) 234

2 4   10 1000

(A) 12.204

(B) 12.024

(C) 12.402

(D) 12.240

2

(B) 0.006 (D) 0.06

1  10

(A) 2.1 (C) 2.001 6.

(B) 0.2 (D) 0.25

3  5 (A) 0.6 (C) 0.0006

5.

(B) 1.1 (D) 1.001

5  2 (A) 0.5 (C) 2.5

4.

(B) 2.2 (D) 2.002

1  10

(A) 0.11 (C) 1.01

(A) 111.01

0.005 =

1 20

12. 12 kg 20 g =

1 111   100

12 

(B)

11. 5 kg 5 kg =

1.44 = (A)

1 2

10. 5g =

0.02 = (A)

(A)

(B) 2.01 (D) 2.0002

Between which two whole numbers on the number line is the number 5.3 lies? (A) 1 and 2

(B) 2 and 3

(C) 3 and 4

(D) 5 and 6

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6th Class Mathematics

54 7.

8.

10  2 

1 2   10 100

13. Express the following in decimal form:

(A) 12.12

(B) 12.21

(C) 11.11

(D) 21.12

(A) 0.07 (C) 0.007

(A) 1.5 kg

(B) 1.05 kg

(C) 1.005 kg

(D) 1.0005 kg

Which of the following has same values (A)

3 4

(C) 0.75

(A) 0.5 (C) 0.125

15 ? 20

75 100 1 (D) 2 (B)

10. Which of the following fraction has same value as 0.005? (A)

5 100

1 (C) 200

(B)

5 1000

1 (D) 20

11. Which of the following is equivalent to the decimal 2.25? (A) 1.5 × 1.5 (B) 0.3 × 5 × 1.5 (C) 0.9 × 2.5 (D) 225 × 0.01 12. Which of the following are correct? (A) The decimal value of

6 is 1.44 25

2 4   12.204 (B) 12  10 1000 (C)

(B) 0.7 (D) 0.0007

14. Change the following fractions into decimals:

1 kg 500 g =

II. Multiple Correct Answer Type 9.

1 12  2 2 4

(D) 1.5 × 1.5 = 2.25 Use the following information to answer the questions given below. Changing fractions to decimals and vice versa I. To change a fraction to a decimal: Divide the numerator by its denominator. II. To change a decimal to a fraction: (A) Count the number of digits in the decimal. (B) Then, convert into an equivalent fraction with a denominator that is a multiple of 10. (C) Simplify the answer to the lowest terms whenever possible.

1 8

(B) 0.25 (D) 0.0625

15. Express the following in decimal form: 5

65 1, 000

(A) 6.025 (B) 5.65 (C) 5.065 (D) 6.0625 16. Convert the following decimal into fraction : 1.228 (A) 1

1000 228

(B) 1

58 120

228 57 (D) 1 1000 250 17. Change the following decimals to fractions: 9.45 (C) 1

(A) 9 (C)

45 100

945 1000

85 (B) 11 100 (D) 9

0.45 100

IV. Integer Type 18. 0.17 + ? = 5.17. 19. 8244  ? = 1374. 20. 10.875 – ? = 1.875. 21. 6.25  2.5 = 5  ? .

V. Matrix Matching Column I

ColumnII

(A) 1.5

(p)

(B) 1.23

(q) 1

(C) 0.625

(r) 1

(D) 1.42

(s) 9

III. Paragraph Type

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7 10

625 1000

(t) 1

42 100 23

100

45 100

5 10

Number System-II

55

Squares and Square Roots 1. Squares of Numbers 1.

If the estimated answer is in the range, then it is a reasonable answer.

The square of a number is the product of a number multiplied by itself. Example:

2. Determining the Squares of Numbers using a calculator

The square of 5 is 5 × 5 = 25. The square of

2.

1 1 1 1 is × = . 3 3 3 9

The square of a number can be determined using a calculator

3. Perfect Squares

The square of – 0.4 is (–0.4) × (–0.4) = 0.16.

1.

Perfect squares are the squares of whole numbers.

The square of a number is written using the square notation.

2.

Perfect squares are formed by multiplying a whole number by itself.

Eg: (–2) × (–2) is written as (–2)2.

Example:

7 × 7 is written as 72.

1 1 × is written as 8 8 3. 4. 2. 1.

2.

1=1×1

4=2×2

1   . 8

Perfect squares

Determining if a number is a perfect square

62 is read as ‘6 squared’ or ‘the square of 6’ or ‘6 to the power of 2’. The square notation is expanded to obtain the product of the squared number.

1.

A number is a perfect square when it is a product of the same two whole numbers.

2.

To determine whether a given number is a perfect square, find all the prime factors of that number.

3.

Any number which is expressible as pairs of equal factors is a perfect square.

Determining the squares of numbers without using calculator

The square of a number is calculated by multiplying the number by itself. Example: 72 = 7 × 7 = 49

Examples:

Before calculating the square of a mixed number, change the mixed number into an improper fraction. Number Square of number Answer

225 = (5  5)  (3  3)

0. 2

9=3×3

2

 225 is a perfect square.

4.

Square Roots Square roots of numbers

0.22

0.04 

1 d.p.

2. d.p

0.02 

The square root of a positive number is a number which, when multiplied by itself, equals the given positive number.

0.022

0. 0004 

2 d.p.

Example: What is the square root of 25?

4 d.p.

0. 002 

0.0022

The square root of 25 is 5 because when 5 is multiplied by itself, equals 25.

0.000 004  

3 d.p.

6 d.p.

5 × 5 = 25

1.



Estimating the squares of numbers To estimate the square of a number: Method 1: Round off the number to the nearest whole number. Then, calculate the square of that number. Method 2: Determine the range of the square of that number.

2.

The square root of 25 is 5.

The symbol for square root is

.

Example: The square root of 25 is denoted as 

25 .

25 = 5 www.betoppers.com

6th Class Mathematics

56

25 is read as ‘the square root of twenty

(A)

13  13  13  13  169 = 13

Determining the square roots of positive numbers without using a calculator

(B)

7 7 77 49 7    = 11 11 1111 121 11

The square root of a fraction is determined by finding the square root of the numerator and denominator separately.

If x is a positive number, then

five’.

1.

x x  Example: 2.

16 = 64

4 1 16 = = 2 64 8

Example:

Some fractions must be reduced to fractions with perfect squares as their numerators and denominators before their square roots can be calculated.

3.

 10 

2

= 10

8  9  8  9  72

Estimating the Square Roots of Numbers Estimate the square root of a number by determining the range of the square root of that number.

Example:

Finding Square roots of perfect Squares

I.

17 81 81 9 1    1 64 64 8 64 8

The square root of certain decimals are obtained by first changing the decimals into fractions with perfect squares as their numerators and denominators.

Example:

24 3.24  3 100 25

 (Write 3.24 in the form of a fraciton

i.e., 3

=

24 ) 100

81 6  (Change 3 25 into an improper 25

fraction) =

9 = 1.8. 5

Multiplying two Square Roots

By Long Division Method If we have to find out the square roots of very large numbers, the method of finding their square roots by prime factorization becomes very lengthy, difficult and also time consuming. So, we use long division method. Before we give examples to establish and illustrate the algorithm method, let us discuss the pattern of digits in a number and their number of digits in its square root. Study the following table carefully.

6

1.

10  10 

To find the square root of a mixed number, first change the mixed number into an improper fraction.

1 4.

x

x  y  xy Example:

50 25 25 5    49 98 49 49 7

2

If x and y are positive numbers, then

25

Example:

 x

No.of digits in a perfect square number 1 or 2 2 or 4 5 or 6

No.of digits in the square root of the number 1 2 3

and so on. It is clear from the above table that: (i)

If there are 2 or less than 2 digits in a perfect square, then the square root of that number will have one digit.

The product of a square root, when multiplied by itself, results in the number itself.

(ii) If a number consists of either 3 or 4 digits, then its square root will have 2 digits.

Example:

(iii) If a number consists of either 5 or 6 digits, then its square root will have 3 digits. We coincide that,

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Number System-II (i)

57 Method:

If the number of digits in a perfect square is even, the number of digits in its square root is half of the number of digits in the number.

When the number is given in decimal form, we start pairing of numbers on the left and right of the decimal point. In the decimal part, digits are paired from left to right starting with the tenths place. In case the decimal portion of any number of the last digit is left unpaired, make it an even number of decimal places by putting zeros on the right side of that digit if necessary. Now mark off periods and find the square root by long division method. Put the decimal point in the square root as soon as the integral part is exhausted.

(ii) If the number of digits in a perfect square is odd, the number of digits in its square root is half of the number obtained by adding 1 to the number of digits in the number. Note: When we want to find the number of digits in the square root of a number, place a bar over every pair of digits starting with the units digit and count the number of these bars. Thus, the number of bars will be the number of digits in the square root.

Note: The number of digits in the square root is always equal to the number of pairs in the given square number.

For example: Square root of 2.25 will have two digits (2 will be called the first period and 25 will be called the second period.

Formative Worksheet 1.

Square root of 11.56 will have two digits (11 is the first period and 56 will be the second period

(A) 9

Square root of 29.8116 will have three digits (29 is the first period, 81 is the second period, 16 is third period. Steps in Division Method (i)

(D)  2.

Place bars over every pair of digits starting with the units digit. Each pair and remaining one digit (if any) is called period.

(ii) Think of the largest number whose square is less than or equal to the first period. Take the square root of this number as the divisor and the quotient.

(C)

(E) 0.5

(F) –2.7

Write following numbers using the square notation.

 2  2 (B)         5  5 3.

 2 (B)     3

(A) (15) 4.

Repeat steps (ii), (iii) and (iv) till the last period has been taken up.

5.

2

(C) (–7.4)2

Calculate the value of the following without using a calculator. (A) (15)2

(B) (–60)2 2

(C) 0.011

 3 (D)  1   8

2

Estimate the values of the following. (A) (2.893)2

6.

(C) 3.8 × 3.8

Write each of the following numbers in its expanded form. 2

(iv) Now, new divisor is obtained by taking double the quotient and annexing with it a suitable digit which also becomes the next digit of the quotient, chosen in such a way that the product of new divisor and this digit is equal to or just less than the new dividend.

Square roots of Numbers given in Decimal forms

5 8

3 7

(B) – 14

(A) (–13)×(–13)

(iii) Subtract the product of divisor and quotient from first period and bring down the next period to the right of the remainder. This becomes the new dividend.

Now, the quotient thus obtained will be the required square root of the given number.

Write down the square of each of the following numbers in the form of the number multiplied by itself.

(B) (698.6)2

(C) (0.75)2

Estimate the square of each of the following numbers by determining the range in which the value lies.

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58 7.

Use a scientific calculator to find the value of  9 (B)    13 

2

(A) (49) 8. 9.

6.

6th Class Mathematics Simplify the following by factor method: (A). (i)

2

4 9

(ii) 100  81

2

(C) (7.575)

List the first ten perfect squares. Determine whether the following are perfect squares. (A) 484 (B) 49,000

10. Using the square root notation

(iii) 36  4

(iv)

9  16

(v)

(vi)

2500  441

441  121

(vii) 529  64

  , write the

(viii) 10,000  100

square root of the following squares.

(B). (i)

(ii)

64  9

(iii) 100  81

(iv)

729  100

(v)

(vi)

7225  625

49  4

2

121  11  (A) 3 × 3 = 9 (B)   = 144  12  (C) (0.16)2 = 0.0256 11. Calculate the value of (A)

(B)

49

(C)

4

(vii) 1764  1444

82

12. Calculate the values of the following.

144 196

(A)

(B)

3

1 16

(C)

(viii) 8100  7921

20 125

(C). (i)

Conceptive Worksheet 1.

3.

(ii) – 5

(iii) –

5.

(iv) 15625  225

(v)

(vi)

(D). (i)

(v) 3.2 Find the value of each of the following: (i)

36

(ii)

(v)

10.24

64 (iii) 144 (iv) 169

Find the squares of the following by actual multiplication: (i) 0.3 (ii) 0.8 (iii) 1.4 (iv) 2.1

1 5 16 (vii) (viii) 10 10 10 Find two numbers between which the square roots of the following numbers lie: (i) 6 (ii) 12 (iii) 23 (iv) 68 (v) 120 (vi) 190 (vii) 246 (viii) 375 (ix) 862 (x) 1000 (xi) 1100 (xii) 1250 (xiii) 244 (xiv) 315 (xv) 535 (xvi) 340 Find the square root of the following numbers by factor method. (i) 64 (ii) 100 (iii) 144 (iv) 289 (v) 900 (vi) 196 (vii) 529 (viii) 1225 (ix) 1521 (x) 4624 (xi) 7056 (xii) 8100 (xiii) 9216 (xiv) 17424 (xv) 23716 (xvi) 10404 (xvii) 40401 (xviii) 10,000 (xix) 38025 (xx) 20736

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(ii) 100  169

(iii) 324  144 361  100

(vii) 196  9

1 2 (iv) – 2 4 3

576  36

256  4

(viii) 64  169 (ii)

6400  64

(iii)

2601  9

(iv)

784  4

(v)

3844  16

(vi)

5929  121

(vii) 1296  144 (viii) 15876  81 7.

(v) 0.05 (vi) 4.

4  49

Write down the square of each of the following: (i) 12

2.

961  484

Write the number of digits in the square root of the following numbers :(no calculation needed) (i) 324

(ii) 676

(iii) 1369

(iv) 9604

(v) 21904

(vi) 110889

(vii) 4937284 (viii) 9803161 8.

9.

Find the square root of the following numbers by division method. (i) 361 (iv) 8464 (vii) 12321

(ii) 529 (v) 9216 (viii) 13225

(ix) 49729

(x) 96721

(iii) 2304 (vi) 10609

Identify which of the following are perfect squares. 10, 100, 15, 30, 81

Number System-II

59

Summative Worksheet 1.

(–0.3)2 = (A) – 0.09 (C) 0.09

HOTS Worksheet 1.

(B) – 0.9 (D) 0.9

2.

2

2.

 4   =  9

(A)  (C)

3.

16 81

(B) 

16 81

(D)

4.

4 9

1 4

(B)

5.

11 4

25 121 (D) 4 4 2 If (6.453) = 41.64, then (64.53)2 = (A) 416.4 (B) 4,164 (C) 41,640 (D) 4,16,400 The value of (3.42)2 lies between (A) 6 and 7 (B) 9 and 16 (C) 26 and 36 (D) 40 and 44 (5.79)2 rounded off to three decimal places is (A) 33.524 (B) 33.520 (C) 33.5 (D) 33 (C)

4.

5.

6.

7.

8.

9.

6.

7.

480 is (A) greater than 20 (B) less than 20 (C) equal to 20 (D) equal to 40 Calculate the value of

3 (B) 4

1 (C) 1 2

1 (D) 2 2

Calculate the value of

9 5

(B)

3 5

(C)

3 25

(D)

1 25

10. Given that

4.8 = 2.191 and

find the value of (A) 670.89 (C) 67.089

Given that 8.5 = 2.915 and 85 = 9.320. Find

0.00085 .

State whether each of the following statements is true or false. (i) (5.01)2 lies between 16 and 25 (ii) (26.1)2 lies between 400 and 900 (iii) 144 < (12.1)2 < 169 (iv) 0.1 < (0.12)2 < 0.2 (v) 0.0009 < (0.03)2 < 0.0016 (vi) 4,000 < (23.2)2 < 9,000 Find the value of each of the following: (A)

2 3

(B)

3 1  4 4

(C)

1.4  2

(D)

2 1 1  2 3 5

Find the square root of each of the following numbers, without the use of calculator. (A) 16

8.

16 1 25

(A)

430000 .

(B) 6

1 4

(C)

1 4

8 1 (E) 3 50 16 Find the value of each of the following, without the use of calculator. (D)

1 6 . 4

1 (A) 2

4.3 = 2.074 and 43 = 6.557. Find

the value of

 1 5  =  2

(A)

Given that the value of

4 9

2

3.

The length of a square is 2.54 cm. Find the area of the square correct to two decimal places. The area of a square is 9.86 cm2, what is its length correct to two decimal places?

9.

7 9

(A)

121

(B)

1

(D)

0.64

(E)

243 300

(C)

6.25

Two-thirds of the area of a square is equal to

1 the area of the 8 original square. Find the area of the second square and the length of its side. 10. Two squares of sides 9 cm and 12 cm have 48 cm2. Another square is

48 = 6.928,

4,800 – 480 . (B) 473.7 (D) 47.37

5 of the area of another 9 square. Find the length of the side of this square, giving your answer correct to 2 decimal places. total area equal to

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6th Class Mathematics

60 12.

IIT JEE Worksheet I.

Single Correct Answer Type

1.

The value of

2.

6.9 100 lies between (A) 800 and 900 (B) 200 and 300 (C) 80 and 90 (D) 20 and 30 Given that (3.9268)2 = 15.42, find the value of

(A) 87 13.

154 200 .

(A) 3926.8 (C) 39.268 3.

(B) 392.68 (D) 3.9268

Given that

4.

Given that

(B) 27.83 (D) 3.33

64.52 = 8.032, what is the value

2

of (80.32) ? (A) 64,520 (C) 645.2 5.

Given that

(B) 6,452 (D) 64.52

45 = 6.708 and 45 = 2.121, find

the value of

6.

4,500 +

450 .

(A) 882.9 (B) 458.7 (C) 88.29 (D) 45.87 Which of the following values is not equal to 5? (A)

5

(C)

( 5) 2

(B)

2

 5

14. 15.

2

16. 7.

(D) – (5) 2

Given that (5.142)2 = 26.44, find (A) 5 142 (C) 51.42

8.

The value of

17. 264 400 .

(B) 514.2 (D) 5.142

(B) 54.59 (D) 74.29

10.

Which of the following one false for

94

(A)

9+ 4

(B) 3 + 2

(C)

13

(D) None of these

5 = ____________

(A) 2.3 (upto 1st decimal) (B) 2.24 (upto 2nd decimal) (C) 2.2 (upto 1st decimal) (D) 2.23 (upto 2nd decimal) 11. The square of a number ‘n’ is: (A) (n)2 (B) 2n (C) (n  n) (D) none of these www.betoppers.com

(C)

87 (D) 10

87 100

x 2  y 2  z 2  _________

(A)

x 2  y2  z 2 (B) x  y  z

(C)

 xyz 

2

(D) None of these

Finding Square root of a Number by Prime Factorization Method The square root of a number can be found by prime factorization method using the following steps. Step-1: Write the prime factorisation of the given number. Step-2: Pair the factors such that primes in each pair are equal. Step-3: Choose one prime from each pair and multiply all such primes. Step-4: The product thus obtained is the square root of the given number. Answer the following questions Find the square root of 24336. (A) 56 (B) 156 (C) 256 (D) 356 Find the square root of 729 (A) 13 (B) 17 (C) 27 (D) 37 Find the square root of 8100 (A) 30 (B) 60 (C) 90 (D) 120 Find the square root of 9216 (A) 26 (B) 56 (C) 76 (D) 96

18.

132  36  4 = 19. Given that (y – 4)2 = 16, y =

20. 9 9 = 21. The square root of 16

II. Multiple Correct Answer Type 9.

87 10

IV. Integer Type

2980 correct to two decimal

place is: (A) 17.26 (C) 58.75

(B)

III. Paragraph Type

6.4 = 2.53, find the value of

640  64 . (A) 33.30 (C) 10.53

0.87  __________

V. Matrix Matching Column I (A) Value of

(p)

144  16  4 (B)

7,456 lies

Column II 2 5

(q) 0.75

between (C)

1 52 (D) Calculate the value of 1 – (0.5)2 169  69 

(r)

80 and 90

(s)

75 100

(t) 20 (u) 50 and 60

Ratio & Proportion

Ratio & Proportion

Learning Outcomes

B y t h e e n d o f t h i s c h a p t e r , yo u w i l l un de r s t an d

• • •

1.

Ratio Finding ratio Comparison of Ratios

• • •

Proportion Ratio of Three Quantities Unitary Method

Ratio

2.

A ratio is a relation between two quantities of the same kind. Ratio between the two numbers say 3 and 4 may be written as a (i) (ii) (iii)

3 4 division, 3  4 or with the ratio sign (:), 3 : 4

3 =3  4 = 3 : 4 4

(Read as 3 to 4 or 3 ‘is to’ 4) In the ratio 3 : 4 (i) 3 and 4 are called terms of the ratio (ii) 3 is called the first term or anticedent (iii)4 is called the second term or consequent. Example Varun has 6 balloons and Tarun has 8 balloons (i) We say the ratio of their balloons is 3 : 4 (The common factor is eliminated) (ii) We can compare thier balloons in two ways: Comparison by Difference Tarun has 2(8–6) more balloons than Varun. This method is called comparison by difference. Comparison by Division

6 3 If we divide 6 by 8, we get = (lowest 8 4 term). This method is called comparison by division. Thus the ratio of the balloons of Varun and Tarun is 3 : 4 Other examples: (A) The ratio between Rs. 2 and Rs. 3 = 2 : 3 (B) The ratio between 3 cm and 4 cm = 3 : 4 (C) The ratio between 1 cm and 20 mm = 1 : 2 (since 20 mm = 2 cm) (D) The ratio between 10 kg and 15 kg = 2 : 3 (E) The ratio between 3 kg and 1000 g = 3 : 1 (since 1000 g = 1 kg)

Finding Ratio To find a ratio between two quantities, we must have The quantities of the same kind, for example 2 books can be compared with 5 books. The quantities must be expressed in the same units. Example to compare Rs. 2 with 50 p, the ratio is 200 :50 or 4 : 1. Usually ratio is expressed in its simplest form, i.e. the form in which its terms have no common factor except 1 Example 4 : 8 is expressed as 1 : 2. Multiplying or dividing both terms of a ratio by the same number does not change the value

fraction,

It means,

Chapter -3

61

of the ratio e.g.

5 5 2 10    9 9 2 18

The order of the ratio is also very important, Example the ratio 3 : 7 is different from the ratio 7 : 3.

3.

Comparison of Ratios Let us take the two ratios 3 : 5 and 4 : 7 and compare them 3:5=

3 4 and 4 : 7 = 5 7

Now we compare the two fractions by making their denominators equal L.C.M. of 5 and 7 is 35. 

and since or Hence

3 3 7 21    5 5 7 35 4 4 5 20    7 7 5 35 21 > 20



21 20  35 35

3 4  5 7 3:5 > 4:7

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6th Class Mathematics

62 I.

Equivalent Ratios To determine whether two ratios are equivalent or not, either multiply or divide the numerator and denominator of both parts of one ratio by the same number. The choice of the number will depend on the other ratio to be compared.

II.

4.

Simplify ratios to the Lowest Terms The ratio h : k is in the lowest terms if h and k do not have any common factor and h and k are whole numbers. To reduce a ratio to the lowest terms, divide h and k by their common factor. If h or k is a fraction or a decimal number, then multiply by a suitable factor to make it a whole number.

III.

5.

15 10 = = = 18 6 30 6.

Ratios related to a given Ratio In general, if A : B = x : y then (A) B : A = y : x (B) A : (A + B) = x : (x + y) (C) B : (A + B) = y : (x + y) (D) A : (A – B) = x : (x – y) where x > y where x > y (E) B : (A – B) = y : (x – y) where x > ywhere x > y (f) (A + B) : (A – B) = (x + y) : (x – y)

7.

8.

Formative Worksheet 1.

2.

There are 20 girls and 15 boys in a class. (A) What is the ratio of number of girls to the number of boys ? (B) What is the ratio of number of girls to the total number of students in the class? Out of 30 students in a class, 6 like football, 12 like cricket and remaining like tennis. 9.

3.

Find the ratio of : (A) Number of students liking football to number of students liking tennis. (B) Number of students liking cricket to total number of students. See the figure and find the ratio of : (A) Number of triangles to the number of circles inside the rectangle. (B) Number of squares to all the figures inside the rectangle. (C) Number of circles to all the figures inside the rectangle.

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Distance travelled by Hamid and Akhtar in an hour are 9 km and 12 km. Find the ratio of speed of Hamid to the speed of Akhtar. Fill in the following blanks

10.

[Are these equivalent ratios ?] Find the ratio of the following : (A) 81 to 108 (B) 98 to 63 (C) 33 km to 121 km (D) 30 minutes to 45 minutes In a year, Seema earns Rs. 1,50,000 and saves Rs. 50,000. Find the ratio of : (A) Money that Seema earns to the money she saves. (B) Money that she saves to the money she spends. Out of 1800 students in a school, 750 opted basketball, 800 opted cricket and remaining opted table tennis. If a student can only one game, find the ratio of : (A) Number of students who opted basketball to number of students who opted table tennis. (B) Number of students who opted cricket to the number of students who opted basketball. (C) Number of students who opted basketball to the total number of students. Consider the statement: Ratio of breadth and length of a hall is 2 : 5. Complete the following table that shows some possible breadth and lengths of the hall.

Present age of father is 42 years and his son is 14 years. Find the ratio of : (A) Present age of father to the present age of son. (B) Age of the father to the age of son, when son was 12 years old. (C) Age of father after 10 years to the age of son after 10 years. (D) Age of father to the age of son when father was 30 years old.

Ratio & Proportion

63

Conceptive Worksheet

4.

Proportion

1.

Two boys weigh 25 kg and 20 kg. Find the ratio of their weights.

An equality relation between two ratios is called the proportion. Example:

2.

The volumes of water in two containers are 3 litres and 1,500 ml. Find the ratio of the volumes of water.

(i)

3. 4.

Determine whether the following ratios are equivalent ratios 2 : 3 and 10 : 15. Find the ratio of the following : (A) 30 minutes to 1.5 hours (B) 40 cm to 1.5 m

6.

In a college out of 4320 students, 2300 are girls. Find the ratio of : (A)

Number of girls to the total number of students.

(B)

Number of boys to the number of girls.

(C)

Number of boys to the total number of students.

7.

Cost of a dozen pens is Rs. 180 and cost of 8 ball pens is Rs. 56. Find the ratio of cost of a pen to the cost of a ball pen.

8.

Divide 20 pens between Sheela and Sangeeta in the ratio of 3 : 2.

9.

Mother wants to divide Rs. 36 among her daughters Shreya and Bhoomika in the ratio of their ages. If age of Shreya is 15 years and age of Bhoomika is 12 years, find how much Shreya and Bhoomika will get.

10.

Given A : B = 4 : 3. Find A if A + B = 28.

11.

Given PQR is a straight line and PQ : QR = 3 7. (B) PQ : PR

(C) QR : QR – PQ 12.

13.

14.

1 x  , Here the four quantities 1, 2 3

(ii)

If

(iii)

2, x and 3 are in proportion. If 2, 3, 4 and 6 are in proportion, then

2 4  (or 2 : 3 : : 4 : 6) 3 6

There are 102 teachers in a school of 3300 students. Find ratio of the number of teachers to the number of students.

Find (A) QR : PQ

2 4  , Here the four quantities 2, 3 6

3, 4 and 6 are said to be in proportion.

(C) 55 paise to Re. 1 (D) 500 ml to 2 litre 5.

If

If the ratio of male teachers to female teachers in a school is 2 : 5, find the total number of teachers in the school, if the number of female teachers in the school is 25. A fruit-seller sold p apples and m mangoes. If p : m = 4 : 5 and the sum of the number of the two fruits sold is 36, find the difference between the number of the two fruits sold. 4 exercise books cost Rs. 4.80 and a dozen of the same exercise books cost Rs. 14.40. Is the cost of the exercise books proportional to the number of exercise books?

x 4  3 6

(iv)

If x : 3 : : 4 : 6, then

(v)

If a, b, c and d are in proportion or if a : b = c : d or if

a c  , then ad = bc (by b d

cross multiplication) Also, a and d are called extremes or end terms and b and c are called means or middle terms. We can say that four terms or number are said to be in proportion when the product of extreme terms (end terms) is equal to the product of mean terms (middle terms). In a proportion, if the middle terms are repeated, then each of the middle term is called the mean proportional. For example, in a proportion 3, 9, 9, 27; 9 is called the mean proportional. Product of extremes = 3  27 = 81 Product of means = 9  9 = 81 3, 9, 27 are said to be in proportion and the middle term 9 is called the mean proportional. Hence if a : b = b : c then ac = b2 (product of extremes = product of means) The middle term b is called the means proportional between a and c. Now let us see the other proportions involving the terms of the given proportion. Let us take a proportion 1 : 2 = 4 : 8

1 4 1 and 4 : 8 =  2 8 2

(A)

1:2=

(B)

thus 1 : 2 = 4 : 8 We can also write the above proportion as 2 : 1 = 8 : 4 2:1=

2 2 = 2 and 8 : 4 = = 2 1 1

Thus 2 : 1 = 8 : 4

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6th Class Mathematics

64 (C)

Second way of writing the above proportion is: 1 : 4 = 2 : 8 1:4=

(D)

14.

1 2 1  and 2 : 8 = 4 8 4

Thus 1 : 4 = 2 : 8 Third way of writing the same proportion is: 4 : 1 = 8 : 2 4:1=

15.

1 8 = 4 and 8 : 2 = = 4 4 2

Thus 4 : 1 = 8 : 2 Hence there are three more proportions which can be obtained by just changing the positions of the terms of proportion.

5.

16.

Ratio of Three Quantities Ratios can also be used to compare more than two quantities. Example: The ages of three children are 10 years, 11 years and 13 years. The ratio of their ages is 10 : 11 : 13. Note: The three quantities must be in the same unit. The ratio of three quantities can be simplified: (A) by converting all quantities of different units to the same unit; (B) if some or all the quantities are fractions, multiply all of them by the L.CM. of the denominators of the fractions. If any

Conceptive Worksheet 15.

Simplify: 12 : 24 to the lowest terms.

16.

Given h : k = 4 : 3 and the sum of h and k is 21. Find h. Simplify: 6 : 10 : 14 to the lowest forms. Given a triangle ABC. The ratio of the angles A : B : C = 1 : 3 : 5. Find A : B The weight of 24 similar books is 15 kg. Find the weight of 16 similar books. A house painter mixes yellow paint with blue paint to get green paint. The ratio of the volume of the yellow paint to the volume of the blue paint is 5 : 3. If the difference in volumes of the two paints is 500 ml, find the volume of the green paint. If sum of the sides of a triangle PQR is equal to 36 cm and PQ : QR : PR = 2 : 3 : 4, find the length of the sides of the triangle.

17. 18. 19. 20.

1 2

of them is a mixed number like 3 ,

(C) (D)

change it to an improper fraction first. if the quantities have a common factor, divide all of them by the common factor. if some or all the quantities are decimals, convert all of them to whole numbers by multiplying all of them by a suitable power of 10, i.e., 10,100, 1000, etc.

Formative Worksheet 11.

12.

13.

Determine if the following are in proportion : (A) 15, 45, 40, 120 (B) 33, 121, 9, 96 (C) 24, 28, 36, 48 (D) 32, 48, 70, 21 (E) 4, 6, 8, 12 (F) 33, 44, 75, 100 Write True (T) or False (F) against each of the following statements. (A) 16 : 24 :: 20 : 30 (B) 21 : 6 :: 35 : 10 (C) 12 : 18 :: 28 : 12 (D) 8 : 9 :: 24 : 27 (E) 5.2 : 3.9 :: 3 : 4 (F) 0.9 : 0.36 :: 10 : 4 Are the following statements true ? (A) 40 persons : 200 persons = Rs. 15 : Rs. 75 (B) 7.5 litres : 15 litres = 5 kg : 10 kg (C) 99 kg : 45 kg = Rs. 44 : Rs. 20 (D) 45 km : 60 km = 12 hours : 15 hours.

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Determine if the following ratios form a proportion. Also, write the middle terms and extreme terms where the ratios form a proportion. (A) 25cm : 1m and Rs. 40 : Rs. 160 (B) 39 litres : 65 litres and 6 bottles : 10 bottles (C) 2kg : 80 kg and 25 g : 625 g (D) 200 ml : 2 litre and Rs. 4 : Rs. 50. A total of 700 biscuits are to be distributed to 3 boys Ali, Ram and John. The number of biscuits received by Ali and Ram are in the ratio of 2 : 3 and the number received by John and Ram are in the ratio of 5 : 4. Find the number of biscuits received by each of them. If p : q : r = 1 : 2 : 3 and the difference between q and r is 5, find q + r, q, r and the sum of p, q and r.

21.

6.

Unitary Method When two quantities are related such that an incr ease or decrease in one causes a corresponding increase or decrease in the other. This is called direct proportion. Example: Cost of 2 apples is Rs. 12. What will be the cost of 6 apples? Solution:In solving the problems of this kind, we first find the value of one (unit) and then proceed to find the value of the required quantity. Clearly, cost of 6 apples will be more Cost of 2 apples = Rs. 12 12 = Rs. 6 2  Cost of 6 apples = Rs. 6  6 = Rs. 36

Cost of 1 apple = Rs.

Ratio & Proportion

65 28.

Alternative: Apples 2 6

Cost of Rs. 12 x

2 12  6 x 2x = 6  12

6  12 x= 2

29.

30. 31. 32.

x = Rs. 36 Cost of 6 apples is Rs. 36. The method of finding first the value of one (unit) quantity from the value of given quantities and then finding the value of the required quantities is called the Unitary Method of Method of One. In this method, the first line of statement be arranged in such a way that the quantity which is of same denomination as the answer (unknown quantity) comes last.

Formative Worksheet 17. 18. 19.

20.

21.

22.

23. 24. 25.

26.

27.

If the cost of 7m of cloth is Rs. 294, find the cost of 5m of cloth. Ekta earns Rs. 1500 in 10 days. How much she will earn in 30 days ? If it has rained 276 mm in the last 3 days, how many cm of rain will fall in one full week (7 days) ? Assume that the rain continues to fall at the same rate. Cost of 5 kg of wheat is Rs. 30.50. (A) What will be the cost of 8 kg of wheat ? (B) What quantity of wheat can be purchased in Rs. 61 ? The temperature dropped 15 degree Celsius in the last 30 days. If the rate of temperature drop remains the same, how many degrees will the temperature drop in the next ten days? Shaina pays Rs. 7500 as rent for 3 months. How much does she have to pay for a whole year, if the rent per month remains same ? Cost of 4 dozens of banana is Rs. 60. How many bananas can be purchased for Rs. 12.50. The weight of 72 books is 9kg. What is the weight of 40 such books ? A truck requires 108 litres of diesel for covering a distance of 594 km. How much diesel will be required by the truck to cover a distance of 1650 km ? Raju purchases 10 pens for Rs. 150 and Manish buys 7 pens Rs. 84. Can you say who got the pens cheaper ? Anish made 42 runs in 6 overs and Anup made 63 runs in 7 overs. Who made more runs per over ?

Cost of 20 metres of cloth is Rs. 1500. Find the cost of 12 metres of cloth. A bus covers a distance of 550 km in 5 hours. (i) How far will it go in 7 hours? (ii) How much time will it take to cover a distance of 3080 km? A train takes 6 hours to cover 300 km. How much distance will it cover in 9 hours? The weight of 24 similar books is 15 kg. Find the weight of 16 similar books. A family consumes 135 litres of milk in 45 days. How much milk will it consume in 18 days?

Conceptive Worksheet 22. 23. 24. 25. 26. 27. 28. 29. 30.

31. 32.

33. 34.

If 15 pens cost Rs. 35, what is the cost of 39 pens? The cost of 1 book is Rs. 9. How much will 4 books cost? If the cost of 30 pencils is Rs. 240, what is the cost of 18 pencils? If 3 scales cost Rs. 18. Find the cost of 1 scale. The cost of 5 balls is Rs. 40. What is the cost 10 balls? The cost of 5 books is Rs. 40. What is the cost of 7 books? The cost of 3 bottles is Rs. 69. Find the price of 8 such bottles. If the cost of 5 ice creams is Rs. 15, find the cost of 9 ice creams. A truck runs 495 km on 18 litres of diesel. How many kilometres can it run on 34 litres of diesel? 12 men can reap a field in 25 days. In how many days can 20 men reap the same field. In an army camp, there were provisions for 425 men for 30 days. However, 375 men attended the camp. How long did the provisions last? If a car runs 320 km on 5 litre of petrol, how much petrol will be needed for 160 km run? 18 buckets hold as many as 1728 glasses of water. How many buckets would be needed to hold 1344 glasses?

Summative Worksheet 1.

2.

There are 30 boys and 20 girls in a class. The ratio of the number of girls to the number of boys is (A) 2 : 3 (B) 3 : 2 (C) 2 : 5 (D) 3 : 5 There are 25 boys and 25 girls in a class. The ratio of the number of boys to the total number of students is (A) 1 : 2 (B) 1 : 3 (C) 2 : 3 (D) 3 : 2

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6th Class Mathematics

66 3.

4.

5.

6.

7.

8.

9.

10.

11.

The height of Apala is 150 cm. The height of Pari is 120 cm. The ratio of the height of Apala to the height of Pari is (A) 4 : 5 (B) 5 : 4 (C) 5 : 2 (D) 4 : 1 The cost of a car is Rs. 3,00,000. The cost of a motorbike is Rs. 50,000. The ratio of the cost of motorbike to the cost of car is (A) 1 : 6 (B) 1 : 5 (C) 1 : 4 (D) 1 : 3 The speed of Shubham is 6 km per hour. The speed of Yash is 2 km per hour. The ratio of the speed of Shubham to the speed of Yash is (A) 2 : 3 (B) 3 : 1 (C) 1 : 3 (D) 3 : 2 The length and breadth of a rectangular park are 50 m and 40 m respectively. Find the ratio of the length to the breadth of the park. (A) 4 : 5 (B) 4 : 1 (C) 5 : 1 (D) 5: 4 The ratio 40 cm to 1 m is (A) 2 : 5 (B) 3 : 5 (C) 4 : 5 (D) 5 : 2 In a family, there are 8 males and 4 females. The ratio of the number of females to the number of males is (A) 1 : 2 (B) 1 : 4 (C) 1 : 8 (D) 2 : 1 Which of the following ratio is equivalent to 2 : 3? (A) 4 : 8 (B) 4 : 9 (C) 6 : 9 (D) 6 :12 Which of the following ratio is not equivalent to 10 : 5 ? (A) 1 : 2 (B) 2 : 1 (C) 20 : 10 (D) 30 : 15 Find the ratio of number of circles and number of squares inside the following rectangle:

15.

16.

17.

18.

19.

20.

21.

22.

23.

12.

13.

14.

(A) 3 : 1 (B) 2 : 1 (C) 2 : 3 (D) 3 : 2 There are 20 teachers in a school of 500 students. The ratio of the number of teachers to the number of students is (A) 1 : 20 (B) 1 : 50 (C) 1 : 25 (D) 25 : 1 The ratio of 25 minutes to 1 hour is (A) 7 : 5 (B) 5 : 12 (C) 12 : 5 (D) 5: 1 Out of 30 students in a class, 20 like cricket and 10 like Hockey. The ratio of the number of students liking Hockey to the total number of students is (A) 3 : 1 (B) 1 : 3 (C) 2 : 3 (D) 1 : 2

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24.

25.

26.

The cost of 1 dozen bananas is Rs. 30. The cost of 6 oranges is Rs. 18. The ratio of the cost of a banana to the cost of an orange is (A) 3 : 2 (B) 2 : 3 (C) 6 : 5 (D) 5 : 6 The present age of Hari Kishan is 60 years. The present age of Manish is 30 years. The ratio of the age of Manish to the age of Hari Kishan 10 years ago was (A) 2 : 5 (B) 5 : 2 (C) 2 : 3 (D) 3 : 2 100 students appeared in annual exami­nation. 60 students passed. The ratio of the number of students who failed to the total number of students is (A) 5 : 2 (B) 2 : 5 (C) 2 : 3 (D) 3 : 2 Rs. 100 are divided between Sangeeta and Manish in the ratio 4:1. Find the amount Sangeeta gets. (A) Rs. 80 (B) Rs. 20 (C) Rs. 60 (D) Rs. 50 Which of the following are in proportion? (A) 2, 3, 20, 30 (B) 3, 4, 15, 18 (C) 1, 3, 11, 22 (D) 2, 5, 40, 80 Which of the following is true? (A) 15 : 40 :: 10 : 30 (B) 16 : 48 :: 25 : 75 (C) 40 : 60 :: 30 : 40 (D) 20 : 100 :: 30 : 120 Which of the following is false? (A) 25 g : 30 g :: 40 kg : 48 kg (B) 81 : 91 :: 24h : 27h (C) 32 m : 40 m :: 6 minutes :12minutes (D) 25 km : 60 km :: Rs. 10 : Rs. 24. Which of the following statement is not true? (A) 4 : 7 = 5 : 9 (B) Rs. 5 : Rs. 25 = 12 g : 60 g (C) 30 : 80 = 6 : 16 (D) 12 : 36 = 14 : 42. The cost of 10 notebooks is Rs. 100. The cost of 1 notebook is (A) Rs. 10 (B) Rs. 100 (C) Rs. 20 (D) Rs. 5 The cost of 1 dozen pens is Rs. 24. Find the cost of 30 pens. (A) Rs. 40 (B) Rs. 45 (C) Rs. 30 (D) Rs. 60 The cost of 3 envelopes is Rs. 15. The cost of 10 envelopes is (A) Rs. 20 (B) Rs. 30 (C) Rs. 45 (D) Rs. 50 The cost of 5 kg of tomatoes is Rs. 100. The cost of 2 kg of tomatoes is (A) Rs. 20 (B) Rs. 40 (C) Rs. 30 (D) Rs. 50

Ratio & Proportion 27.

28.

29.

30.

31.

32.

67

The cost of 20 m of cloth is Rs. 400. The cost of 15 m of cloth is (A) Rs. 100 (B) Rs. 200 (C) Rs. 300 (D) Rs. 360 The salary of a month of an employee is Rs. 4000. The annual salary of the employee is (A) Rs. 48000 (B) Rs. 24000 (C) Rs. 12000 (D) Rs. 8000 An aeroplane covers a distance of 5000 km in 5 hours. How much distance will it cover in 2 hours? (A) 1000 km (B) 2000 km (C) 3000 km (D) 4000 km The fare for 5 tickets from Korikalam to Mathura is Rs. 150. The fare for 3 tickets is (A) Rs. 90 (B) Rs. 60 (C) Rs. 75 (D) Rs. 45 150 kg of oil can be filled in 10 containers. To fill 750 kg of oil, how many containers will be required? (A) 10 (B) 20 (C) 40 (D) 50 The cost of 8 almirahs is Rs. 8000. The cost of 1 almirah is (A) Rs. 1000 (B) Rs. 2000 (C) Rs. 4000 (D) Rs. 6000

8.

9.

10.

2.

3. 4.

5.

6.

7.

The area of square ABCD is 144 cm2 and the area of square PQRS is 81 cm2, find the ratio of the side AB to the side PQ. The area of square DEFG is 64 cm2 and the area of square WXYZ is 36 cm2, find the ratio of the perimeter of square DEFG to the perimeter of square WXYZ. The cost of 3 pairs of slippers is Rs. 30. Find the cost of 12 pairs of the same type of slippers. P

E

11.

12.

13.

14.

15.

R

Q

The length of the straight line PQR is 22 cm. If PQ : QR m= 4 : 7, find the length of PQ. The tax on the income of a teacher is Rs. 360. If the income of the teacher is Rs. 24,000, find the ratio of the tax to the income. A school has 600 students. The number of boys in the school is 360. Find the ratio of the number of girls to the number of boys in the school. P

Q

S

R

The perimeter of the rectangle PQRS as shown above is 80 cm. If PQ : QR = 5 : 3, find the length of PQ.

D

F

HOTS Worksheet 1.

The prices of three types of soaps P, Q and R are respectively 80 paise, Rs. 1.20 and 90 paise. Find the ratio of the price of P to the price of Q to the price of R. The prices of three types of shirts A, B and C are respectively in the ratio of 3 : 6 : 5. If the total price of the three types of shirts is Rs. 98, what is the price of short C?

16.

17.

The ratio of  D to E to F is 8 : 4 : 3. Find the measure of D . The price of 3 kg grapes is Rs. 24. The price of 7 kg of the same grapes is: (A) Rs. 8 (B) Rs. 56 (C) Rs. 21 (D) Rs. 36 PQRS is a straight line. PQ = 2 cm and PS = 40 cm. If PR : RS = 7 : 3, then QR in cm is: (A) 12 (B) 10 (C) 26 (D) 28 A sum of money is divided into three parts in the ratio of 2 : 3 : 4. The largest part is Rs. 140. The total sum of money is: (A) Rs. 540 (B) Rs. 315 (C) Rs. 310 (D) Rs. 360 Triangle PQR has sides PQ, QR and PR in the ratio of 4 : 3 : 6. PR is longer than PQ by 6 cm. The sum of the lengths of PR, QR and PQ is: (A) 29 cm (B) 10 cm (C) 20 cm (D) 39 cm A total of 500 pieces of sweets is to be divided among Radha, Ahmed and Krishna respectively in the ratio of 3 : 8 : 14. How many pieces of sweets will Radha get? (A) 60 (B) 80 (C) 160 (D) 280 The ratio of the number of red marbles to the number of blue marbles to the number of white marbles in a box is 5 : 3 : 7. If the number of white marbles in the box is 42, find the total number of marbles in the box. (A) 60 (B) 70 (C) 80 (D) 90 A box contains three types of sweets: chocolate, lemon and barley. Their numbers in the box are respectively in the ratio of 5 : 8 : 3. If the number of lemon sweets is 12 more than the number of chocolate sweets, what is the number of barley sweets in the box? (A) 10 (B) 11 (C) 12 (D) 15 www.betoppers.com

6th Class Mathematics

68 18.

19.

20.

A factory has 28 American workers, 20 Chinese workers and the rest are Indian workers. If the ratio of the number of Chinese workers to the number of Indian workers is 5 : 3, then the ratio of the number of American workers to the Chinese workers to the Indian workers is: (A) 4 : 5 : 3 (B) 7 : 5 : 3 (C) 4 : 5 : 7 (D) 1 : 2 : 3 A certain amount of marbles are given to three students P, Q and R in the ratio 3 : 4 : 5. If P and Q receive a total of 315 marbles, how much does R receive? (A) 135 (B) 225 (C) 505 (D) 625 If

22.

23.

24.

25.

26.

If

2 of the property costs Rs. 1200, then what 3

is the cost of the remaining property? (A) Rs. 1,200 (B) Rs. 600 (C) Rs. 900 (D) Rs. 1,800 27.

If

29.

30.

4 th of strength of a class is 80. What is 5

the total strength of class? (A) 20 (B) 100 (C) 110 (D) 90

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1 hrs? 2

(A) 320 km (B) 350 km (C) 340 km (D) none of these If there are 4000 oranges in 50 boxes and the cost of 400 oranges is Rs. 120. What is the cost of boxes containing 4000 oranges? (A) Rs. 1,600 (B) Rs. 1,200 (C) Rs. 1,800 (D) Rs. 4,800 If

1 2 rd and rd of the property costs Rs. 600 3 3

and Rs. 1200 respectively. What is the cost of total property? (A) Rs. 1,200 (B) Rs. 1,500 (C) Rs. 1,800 (D) Rs. 2,400

1 of it? 2

(A) Rs. 16,000 (B) Rs. 20,000 (C) Rs. 14,000 (D) Rs. 10,000 If 50 kg of wheat cost Rs. 550 what is the cost of 11 kg? (A) Rs. 150 (B) Rs. 121 (C) Rs. 125 (D) Rs. 110 If Ramya types 180 words in 4 minutes. How many words will she type in 9 minutes? (A) 405 (B) 450 (C) 540 (D) 504 If there are 4000 pineapples in 50 boxes. How many pineapples will be there in 35 boxes? (A) 2,000 (B) 2,700 (C) 3,500 (D) 2,800 If 10 cans contain 500 litres of milk. Then in how many cans 200 litres of milk can be filled? (A) 5 (B) 1 (C) 2 (D) 4 1500 sheets are required to make 100 notebooks. How many sheets will be required to make 12 note books? (A) 180 (B) 200 (C) 120 (D) 100

If a train covers 200 km in 5 hrs. How many km will it cover in 8

3 of the property costs Rs. 15,000 what is 4

the cost of

21.

28.

IIT JEE Worksheet I.

Single Correct Answer Type

1.

The ratio of Ali’s age to that of his younger brother is 7 : 5. If his younger brother is 15 years old, then Ali’s age in years is: (A) 18 (B) 17 (C) 21 (D) 28 A

2. C

3.

4.

B

ABC is a triangle. AB = 6 cm. The ratio AB : BC : CA = 2 : 3 : 5. CA is longer than BC by: (A) 3 cm (B) 5 cm (C) 6 cm (D) 2 cm The number of chickens in the three farms P, Q and R are in the ratio of 2 : 3 : 4. The total number of chickens in the three farms is 270 chickens. Farm R sells 20 chickens to farm P and 10 chickens to farm Q. After this sale, what is the ratio of chickens in farms P : Q : R? (A) 8 : 9 : 10 (B) 9 : 10 : 8 (C) 8 : 10 : 9 (D) 10 : 9 : 8 A total of 280 marbles is to be divided among three boys Karan, Rahim and Kiran respectively in the ratio 2 : 5 : 7. Find the difference in the amount between Kiran and Karan? (A) 20 (B) 40 (C) 80 (D) 100

Ratio & Proportion 5.

6.

7.

69

A box contains two types of fruits, apples and oranges. If the ratio of the number of apples to the number of oranges is 3 : 5 and the number of apples in the box is 21, how many oranges are there in the box? (A) 56 (B) 35 (C) 21 (D) 15 The age of Tarun is 20 years and the age of Chandu is 36 years. If the ratio of the age of Chandu to the age of Leela is 3 : 2, then the ratio of the age of Tarun to the age of Leela is: (A) 1 : 2 (B) 2 : 3 (C) 3 : 4 (D) 5 : 6 P

P

13.

14.

R

Q

The diagram above shows a straight line PR.Q is a point on PR such that PQ : PR = 2 : 5. If the length PR is 20 cm, then the length of PQ is: (A) 2 cm (B) 4 cm (C) 8 cm (D) 12 cm 8.

12.

15.

III. Paragraph Type

Q

16. S

9.

10.

II. 11.

R

In the diagram above, the small rectangles in the rectangle PQRS are identical. What is the ratio of the shaded portions to the unshaded portions? (A) 3 : 5 (B) 2 : 1 (C) 2 :5 (D) 3 : 7 The marbles in a box are taken out and put into three bags in the ratio of 3: 2 : 5. If the number of marbles in the box is 150, find the number of marbles in the bag which contains the most number of marbles. (A) 50 (B) 75 (C) 100 (D) 125 The flour in a bag is divided into three portions according to the ratio of 4 : 6 : 9. If the smallest portion is 28 kg, how much is the biggest portion? (A) 54 (B) 63 (C) 72 (D) 98

Multiple Correct Answer Type Which of the following ratios are equivalent to 2 : 3 : 6? (A) 4 : 6 : 18

1 3 1 (B) : :1 2 4 2

(C) 0.4 : 0.6 : 1.2

(D) 12 : 24 : 36

Which of the following vary directly? (A) Speed of a vehicle and time taken to cover a fixed distance (B) Number of days worked and amount of earning (C) Number of books and their price (D) Number of units of current used and amount of charge for consumption Which of the following are in proportion: (A) 1 : 2 ; 3 : 4 (B) 1 : 2 ; 2 : 4 (C) 1 : 2 ; 3 : 6 (D) 1 : 2 ; 2 : 3 3, 9, 15, 5 are not in proportion. But by changing their order they can be in proportion. Find out the correct order to get proportion. (A) 15, 9, 5, 3 (B) 3, 5, 9, 15 (C) 3, 15, 9, 5 (D) 9, 15, 5, 3 The ratio equivalent to 2 : 3 is: (A) 6 : 9 (B) 4 : 6 (C) 8 : 10 (D) 10 : 15

17.

18.

19.

At a school day function there were 125 girls and 100 boys present. Based on this information answer the questions given below. Find the ratio of girls to boys. (A)

4 5

(B)

5 4

(C)

15 4

(D)

4 15

Find the ratio of men to the total number of students present (A)

4 9

(B)

9 4

(C)

3 4

(D)

4 3

Write an equivalent fraction for 5 boys. (A) 2 : 5 (B) 3 : 9 (C) 8 : 10 (D) 10 : 8 Find the ratio of girls to the total number of students present. (A)

3 5

(B)

5 3

(C)

9 5

(D)

5 9

IV. Integer Type 20.

21.

The cost of a pen is Rs. 10. The cost of a pencil is Rs. 2. How many times of the cost of a pencil is the cost of a pen? The monthly salary of Hari Kishan is Rs. 80000. The monthly salary of Manish is Rs. 40000. How many times of the salary of Manish is the salary of Hari Kishan?

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6th Class Mathematics

70 22.

23.

V.

A car requires 5 litres of petrol to cover 80 km. How many litres of petrol are required to cover 32 km? The weight of 50 books is 10 kg. The weight of 25 books is ___________kg.

Matrix Matching

24.

Column I

Column II

(A)

5 Rs. to 50 paise

(p)

20 : 1

(B)

15 kg to 210 g

(q)

10 : 1.

(C)

9 m to to 27 cm

(r)

500: 7

(D)

30 days to 36 hours

(s)

100:3

(t)

72 : 1

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Algebra

Learning Outcomes 

Variables



Rules of arithmetic



Equations



Algebraic Expression

1. Introduction Our study so far has been with numbers and shapes. We have learnt numbers, operations on numbers and properties of numbers. We applied our knowledge of numbers to various problems in our life. The branch of mathematics in which we studied numbers is arithmetic. We have also learnt about figures in two and three dimensions and their properties. The branch of mathematics in which we studied shapes is Geometry. Now we begin the study of another branch of mathematics. It is called Algebra. The main feature of the new branch which we are going to study is the use of letters. Use of letters will allow us to write rules and formulas in a general way. By using letters, we can talk about any number and not just a particular number. Secondly, letters may stand for unknown quantities. By learning methods of determining unknowns, we develop powerful tools for solving puzzles and many problems from daily life. Thirdly, since letters stand for numbers, operations can be performed on them as on numbers. This leads to the study of algebraic expressions and their properties.

2. Variables An unknown quantity can be represented by a variable. Usually, a variable is any letter from the English alphabet that represents an unknown quantity. The relation between the unknown quantity and other quantities can be expressed with the help of the variable. The value of the variable varies with the given condition on the variable. A quantity whose value does not vary is called a constant. An expression consisting of variables, constants and mathematical operators is called an algebraic expression. Mathematical operations such as addition, subtraction, multiplication and division can be easily performed on variables. We can use variables to form expressions based on patterns.

Chapter – 4

By the end of this chapter, you will be understand

The following are some branches of mathematics:

•

The branch of mathematics where letters are used along with numbers is called algebra.

•

The branch of mathematics that deals with numbers, operations on numbers and properties of numbers is called arithmetic.

•

The branch of mathematics that deals with the figures and shapes is called geometry.

Use of Variables Variables are used to frame rules to find the perimeter of a polygon. The perimeter of a polygon can be obtained by adding the lengths of its sides. The following are simple rules to frame the perimeter of geometrical figures using variables. If the length of the side is denoted by variable's', then the perimeter of a square is equal to 4s

If the length and breadth of a rectangle is l, b. b Then a perimeter of a of rectangle is 2 (l + b) If the lengths of the sides of a triangle are denoted by x, y and z then the perimeter of the triangle is equal to x + y + z.

s s

s

s square l b l rectangle

x

y

z tirangle

6th Class Mathematics

72

•

3. Rules of Arithmetic The following are some simple rules for the properties of numbers using variables. Commutative Property of Addition: This property states that two numbers can be added in any order. If a and b represent any two numbers, then a + b = b + a Commutative Property of Multiplication: This property states that two numbers can be multiplied in any order. If a and b represent any two numbers, then a × b = b × a Associative Property of Addition: This property states that three numbers can be added in any order. If a, b and c represent any three numbers, then (a + b) + c = a + (b + c) Associative Property of Multiplication: This property states that three numbers can be multiplied in any order. If a, b and c represent any three numbers, then (a × b) × c = a × (b × c) Distributive Property of Multiplication over Addition: This property states that if a, b, and c represent any three numbers, then a × ( b + c) = a ×b+a×c

An equation that does not have any variable is called a numerical or an arithmetic equation. Eg: 17 × 2 = 34 Different numerical values for the variable are substituted in an algebraic equation, and the solution is obtained by using a method called the trial and error method. If there is no sign of equality between the LHS and the RHS, then it is not an equation. Eg: x  5  6,

Formative Worksheet 1.

• •

The statement “five less than half of a number x” is equivalent to the expression (A)

2.

3.

4. Equations A mathematical statement that indicates that the value of the LHS is equal to the value of the RHS is called an equation.

n  7 are not equations. 9

x 5 2

x 5 2

(C)

(B) n – 7

(C) n × 7

(D) n  7 The algebraic expression for the statement “ten more than thrice a number x” is

An equation puts a condition on the variable. The value for which the equation is satisfied is the solution of the equation.

n  15 is satisfied for Eg: The equation 3 n = 45.

•

The value of the variable in an equation that satisfies the equation, or makes its LHS equal to its RHS, is the solution.

•

An equation can contain numbers and variables.

(A) 10x + 3 (B) 3x + 10 (C) 5.

6.

Eg: a – 2 = 30, 72  9  8 .

•

An equation is said to be algebraic equation if it consists of a variable. Eg: 20x = 400.

•

A single variable equation will have a unique solution. Eg: 15n = 225

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7.

x x  2 (D)  2 5 5

The statement “four more than nine times of a number x” is equivalent to the expression (A) 9x – 4    (B) 4x – 9 (C) 9x + 4   (D) 4x + 9 In a basketball match, Jeff scored n points and Bill scored seven points less than Jeff. How many points did Bill score? (A) n + 7

4.

(B)

x x  3 (D)  10 10 3

Two friends Phil and Steve participated in a chocolate eating competition. Phil ate five chocolates less than thrice the number of chocolates eaten by Steve. If Steve ate C chocolates, then which of the following expressions is equivalent to the number of chocolates eaten by Phil? (A) 5C + 3 (B) 3C + 5 (C) 5C – 3 (D) 3C – 5 What is the value of variable x in the equation 36 + 4x = 48 ? (A) 2 (B) 3 (C) 4 (D) 5 What is the value of variable b in the equation 2b – 11 = 15? (A) 5 (B) 7 (C) 13 (D) 15

Algebra 8.

9.

73

If the perimeter, p of a rectangle with length l and 4. width w is given by the equation p = 2l + 2w, then (A) p and l are constants, while 2 and w are variables (B) 2 and w are constants, while p and l are variables (C) p, w and l are constants, while 2 is a variable (D) 2 is a constant, while p, l, and w are variables The perimeter of a rectangle of length l and 5. width w is given by the formula p = 2(l + w) The variables in the given formula are (A) l, w, and P (B) l, 2, and w (C) 2 and l (D) 2 and w

10. Which of the following expressions ‘correctly’ represents the expression “fifteen divided by a number x”?

15 x (D) x 15 11. Which of the following expressions ‘correctly’ represents the expression “x less than ten”? (A) 15x

(B) x + 15

(C)

x 10 12. Jane has Rs.x with her. The amount with Colin is Rs.15, which is Rs.2 more than that with Jane. Which of the following equations correctly represents the given information? (A) x – 2 = 15 (B) x + 2 = 15 (C) x – 15 = 2 (D) x + 15 = 2 (A) 10 – x (B) x – 10

(C) 10x

2.

3.

7.

8.

In a zoo, the number of monkeys is three more than thrice the number of alligators. If there are y alligators in the zoo, then the number of monkeys is

s s 1 1  4 (B)  4 (C) 4s  (D) 4s  2 2 2 2 What is the value of variable y in  equation 6y – 36 = 12? (A) 3 (B) 6 (C) 8 (D) 9 What is the value of variable a in the equation 45 – 5a = 25? (A) 2 (B) 3 (C) 4 (D) 5 What is the value of variable a in the equation 3a + 5 = 32? (A) 1 (B) 4 (C) 6 (D) 9 The area of a rectangle of length l and width w is given by the formula A = l × w. Which of the following statement is true with respect to the given formula? (A)

(A) There is no constant in the given formula.

(D)

Conceptive Worksheet 1.

6.

The number of shirts with Andy is four more than half the number of shirts with John. If John has s shirts, then which of the following expressions is ‘equivalent’ to the number of shirts with Andy?

(B) There is no variable in the given formula. (C) There is only one constant in the given formula. (D) There is only one variable in the given formula. 9.

Which of the following statements is the appropriate verbal expression for the mathematical expression

5Z  6 ? 2

(A) Two times the difference of a number and six, divided by five (B) Five times the difference of a number and six, divided by two

(C) Six times the difference of a number and five, y y (A) 3y – 3 (B) 3y + 3 (C)  3 (D)  3 divided by two 3 3 (D) Six times the difference of a number and two, Tom’s age is five years less than thrice of Ben’s age. divided by five If Ben is x years old, then Tom’s age, in years, is 10. Which of the following expressions ‘correctly’ (A) 5x – 3 (B) 5x + 3 (C) 3x – 5 (D) 3x + 5 represents the verbal expression “x more than If Sam and Raymond respectively have Rs.x and fifteen”? Rs.(x + 10) in their piggy banks, then which of the following statements is true? x (A) x + 15 (B) 15x (C) (D) x – 15 (A) Sam has Rs.10 more than Raymond. 15 (B) Raymond has Rs.10 more than Sam. (C) Sam has Rs.x less than Raymond. (D) Raymond has Rs.x less than Sam. www.betoppers.com

6th Class Mathematics

74 11. Jay is three years younger than Mike, whose age is y years. Jay’s age, in years, is ‘equivalent’ to which of the following expressions? (A) y – 3 (B) y + 3

(C) y × 3

(D) y  3

12. The statement “three more than four times a number x” is equivalent to which of the following expressions? (A) 4x + 3 (B) 4x – 3 (C) 3x + 4 (D) 3x – 4

way does not affect the value of the expression because addition is commutative that is, you can rearrange things that you are adding and without changing the answer. Example: suppose x = 3. Then the original expression and its rearrangement evaluate as follows: –5x + 2

5. Algebraic Expression A combination of constants and variables connected by the sign of fundamental operations of addition, subtraction, multiplication and division are called algebraic expressions. Eg: 2x3 + x2 – 4x + 5

Algebraic Terms A term in an algebraic expression is any chunk of symbols set off from the rest of the expression by either addition or subtraction. Here are some examples: Expression

Number of Terms

Terms

5x

One

5x

–5x + 2

Two

–5x and 2

Four

z 2 x y, ,  xyz, 3 and 8

x2 y 

z  xyz  8 3

When a term has a variable, it’s called an algebraic term. When it doesn’t have a variable, it is called a constant. For example, look at the following

z expression: x2y + – xyz + 8 3 The first three terms are algebraic terms, and the last term is a constant. As you can see, in algebra, constant is just a fancy word for number.

Rearranging Terms After you understand how to separate an algebraic expression into terms, you can go one step further by rearranging the terms in any order you like. For example, suppose you begin with the expression –5x + 2. You can rearrange the two terms of this expression without changing its value. Notice that each term’s sign stays with that term, though dropping the plus sign at the beginning of an expression is customary. Rearranging terms in this www.betoppers.com

2 – 5x

= –5 (3) + 2

= 2 – 5(3)

= –15 + 2

= 2 – 15

= –13

= –13

Rearranging expressions in this way becomes handy later in this chapter, when you simplify algebraic expressions. As another example, suppose you have this expression: 4x – y + 6 You can rearrange it in a variety of ways: = 6 + 4x – y = –y + 4x + 6 Because the term 4x has no sign, it is positive, so you can write in a plus sign as needed when rearranging terms. As long as each term’s sign stays with that term, rearranging the terms in an expression has no effect on its value. For example, suppose that x = 2 and y = 3. Here is how to evaluate the original expression and the two rearrangements: 4x – y + 6

6 + 4x – y

–y + 4x +6

= 4(2) – 3 + 6

= 6 + 4(2) – 3

= –3 + 4(2) + 6

=8– 3+6

= 6+8–3

= –3 + 8 + 6

=5+6

= 14 – 3

=5+6

= 11

= 11

= 11

Polynomials An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial. Eg: 2x2 – 3x – 5 is a polynomial 1

1 3 x  3x 2  5x 2  x  1 is not a polynomial 2 1

because it contains a term 5x 2 which contains

1 as the power of variable x, which is not a non2 negative integer.

Algebra

75

Degree of Polynomial is one Variable In a polynomial in one variable, the highest power of the variable is called its degree. Eg: 2x + 3 is a polynomial in ‘x’ of degree 1 2x2 – 3x +5 is a polynomial in ‘x’ of degree 2

Formative Worksheet 13. The statement “two times a number y” is equivalent to which of the following expressions?

(A) y + 2 (B) y – 2 (C) y  2 (D) y × 2 5 1 3x  7x  x  is a polynomial in ‘x’of 14. If John has n marbles and loses five of them in a 2 3 game, then the number of marbles left with him is degree 4 equivalent to which of the following expressions? Degree of a Polynomial is two Variables (A) n  5 (B) n × 5 (C) n + 5 (D) n – 5 In a polynomial in more than one variable, the sum of the powers of the variables in each term is 15. The equation 3 + x = 18 can be written in words as computed and the highest sum so obtained is called (A) Three more than a number x equals eighteen the degree of the polynomial. (B) Three less than a number x equals eighteen Eg: 3x4 – 2x3y2 + 7xy3 – 9x +5y + 4 is polynomial in ‘x’ and ‘y’ of degree 5 (C) Eighteen more than a number x equals three 3 2 Degree of x y is 3 + 2 = 5 (D) Eighteen less than a number x equals three Constant Polynomial: A polynomial consisting of 16. Which of the following statements is true about the a constant term only is called a constant polynomial. equation x ÷ 7 = 4? The degree of a constant polynomial is zero. (A) A number x, when divided by 7 equals 4. Eg: 5, 7 (B) 7, when divided by a number x equals 4. Linear Polynomial: A polynomial of degree1 is (C) A number x, when divided by 4 equals 7. called a linear polynomial. (D) 4, when divided by a number x equals 7. 3 1 3 17. If a table costs Rs.x and a chair costs Rs.(2x – 3), Eg: 2  x ,  y , 2 + 3a etc. 4 2 5 then the cost of the chair is 4

2

Quadratic Polynomials: A polynomial of degree 2 (A) Rs.3 more than twice the cost of the table is called a quadratic polynomial (B) Rs.3 less than twice the cost of the table Eg: 2x2 – 3x + 4, 2 – x + x2 are quadratic polynomials (C) Rs.2 more then thrice the cost of the table Cubic Polynomials: A polynomial of degree 3 is (D) Rs.2 less then thrice the cost of the table called a cubic polynomial. 18. Which equation correctly satisfies the statement Eg: x3 – 7x2 + 2x – 3 “twenty-five less than thrice the number ‘n’ equal Bi Quadratic Polynomial: A polynomial of degree six”? 4 is called bi quadratic polynomial. (A) 3n – 25 = 6 (B) 25 – 3n = 6 Eg: 3x4 – 7x3 + x2 – x + 9 (C) 3(n – 25) = 6 (D) 3(25 – n) = 6 A polynomial is said to be a monomial, a binomial or a trinomial accordingly as it contains one term, two 19. Jerry had some chocolates and candies. The number of chocolates was 2 more than twice the number of terms or three terms respectively. Every polynomial candies. The candies were x in number. is an algebraic expression but an algebraic expression need not be a polynomial. The total number of chocolates with Jerry equals Like Terms: Terms having the same literal factors (A) 2x – 2 (B) x + 2 (C) 2x + 2 (D) x – 2 are called like terms. 20. Randy scored x marks in mathematics. His marks in biology were 3 more than two-thirds of the marks 2 obtained by him in mathematics. The marks scored Eg: 2ab, –7ab, ab, 9ab 3 by Randy in biology were (A)

2 3 2 3 x  3 (B) x  3 (C) x  2 (D) x  2 3 2 3 2 www.betoppers.com

6th Class Mathematics

76 21. The number of chairs in an auditorium is 4 more than thrice the number of tables in that auditorium If the number of tables is x, then the number of chairs equals (A) 4x + 3  (B) 3x + 4 (C)

x x  3 (D)  4 4 3

22. John’s pocket money is Rs.2 more than half the pocket money of his sister. If John’s sister gets Rs.x, then John’s pocket money in Rs. is (A) 2x 

1 1 x (B) x  (C)  2 2 2 2

(D) x + 2

23. The number of hours required by a bike to travel a particular distance is twice the number of hours required by a car to travel a particular distance. If the car takes x hours, the number of hours taken by the bike will be (A) (x + 2) (B)

x 2

(C) 2x (D) (x – 2)

24. State the number of terms in the following algebraic expression xy + 2yz + 9? 25. Simplify the following expression: 7m + 5m – 3(–3m)? 26. Simplify the following expression: 5a 2b2 – 4a 2b – (–10a 2 b2 )

Conceptive Worksheet 13. If Ben distributes Rs.x equally among his three children, then each child’s share is equivalent to which of the following expressions? (A) Rs.(x – 3) (B) Rs.(x+3) (C) Rs.(x  3) (D) Rs.(x × 3) 14. The expression “three times a number n” is equivalent to which of the following expressions? (A) n × 3 (B) n + 3   (C) n – 3

(D) n  3 15. Which of the following tables is correct? (A)

x+ 2=5 2 x 3 2 3 x+ 2=5 x

(B)

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Equation

2 3 x+ 2=5 x

(C)

x+ 2=5 2 x 3

(D)

Equation Equation Expression Expression

16. If the length (l) (in m) and width (w) (in m) of a room

1 are related by the expression w  l  1 , then the 2 width of the room is (A) 1 m more than half of the length (B) 1 m less than half of the length (C) 2 m more than half of the length (D) 2 m less than half of the length 17. Which equation correctly satisfies the statement “seven less than eight times the number ‘x’ equals thirty three”? (A) 8x – 7 = 33 (B) 7x – 8 = 33 (C) 7 – 8x = 33

(D) 8 – 7x = 33

18. Which equation correctly represents the statement “the double of the sum of fourteen and twice a number ‘a’ equals forty”? (A) 14 + 2a = 40 (B) 2(14 + 2a) = 40 (C) 2 × 2a + 14 = 40 (D) 2(a + 14) = 40 19. There were x tube lights in an auditorium. The number of fans was 7 less than half the number of tube lights. The number of fans was

x x x x  2 (B)  7 (C)  7 (D)  2 7 2 2 7 20. Mr. Smith’s salary is half of Mr. Gere’s salary. If Mr. Gere’s salary is Rs.x, then the salary of Mr. Smith in dollars is (A)

x (C) x + 2 (D) x – 2 2 21. Angelina weighs 20 pounds less than two-thirds the weight of Rambo. If Rambo weighs x pounds, then Angelina’s weight in pounds is (A) 2x

(B)

Expression

(A)

3 x  20 2

(B)

2 x  20 3

Equation

(C)

2 x  20 3

(D)

3 x  20 2

Expression

Algebra

77

22. The number of flowers in the garden A is 5 less than 7. thrice the number of flowers in the garden B. If the number of flowers in garden B is x, then the number of flowers in garden A is 8.

24. Simplify 2m(5m + 4n) 25. Simplify 2a(3b + 5c + 12a) 26. Raju’s father is thrice as old as Raju. If father’s age is 45 years, then Raju’s age is? 27. Half of a number is added to 18 then the sum is 46. 9. Then the number is ?

2.

The equation 17x + 6 = 40 can be written in verbal form as (A) Six more than forty times a number x equals seventeen (B) Six more than seventeen times a number x equals forty (C) Forty more then six times a number x equals seventeen (D) Forty more than seventeen times a number x equals six The equation x + 7 = 3 can be written as (A) Three more than a number x is seven (B) Three less than a number x is seven (C) Seven more than a number x is three (D) Seven less than a number x is three

3.

5.

6.

(C)

(C) 5x

(D)

x 5

The statement “four more than a number x” is equivalent to which of the following expressions? (C) x × 4

(D) x  4 10. The expression “six less than five times a number n” is equivalent to which of the following expressions? (A) 5n + 6 (B) 6n + 5 (C) 5n – 6 (D) 6n – 5 11. Eddy has three more marbles than twice the number of marbles with Rosy. If Rosy has x marbles, then the number of marbles with Eddy is (A) 3x + 2 (B) 2x + 3 (C) 2x – 3 (D) 3x – 2 12. Which equation correctly represents the statement “seven less than a number ‘x’ when divided by four equals twenty”? (A)

7 x  20 4

(B)

4 x  20 7

(C)

x7  20 4

(D)

x4  20 7

Johnson is three feet shorter than Bryan. 13. Which equation correctly represents the statement “the sum of six and thrice the number ‘n’ when If Bryan’s height is h feet, then Johnson’s height, in divided by five equals twelve”? feet, is (A) h – 3

4.

(A) x + 5 (B) x – 5

(A) x – 4 (B) x + 4

Summative Worksheet 1.

8 (D) 8n n Tom is Sally’s younger brother and is 5 years younger than her. The age of Sally is x years. What is Tom’s age in terms of Sally’s age? (A) n – 8 (B) n + 8

(A) 3x – 5 (B) x – 5 (C) 3x + 5 (D) x + 5 23. Simplify (2a2b – 3bc) – (a2b + 5bc – ca)

Some cadets are marching in a parade. There are 8 cadets in each row. There are n rows of cadets. How many cadets are marching in the parade?

(B) h + 3

(C) 3h

(D)

h 3

(A)

5  3n  12 6

(B)

6  3n  12 5

Eric secured five marks more than twice the marks 3  n  5 3  n  5  12  12 (C) (D) secured by Eddy. If Eddy secured M marks, then 6 5 which of the following expressions is equivalent to 14. The weight of a chair is x kg and the weight of a the marks scored by Eric? table is 8 kg lesser than twice the weight of the chair. (A) 2M + 5 (B) 2M – 5 (C) 5M + 5 (D) 5M – 5 What is the weight of the table in terms of x? What is the value of variable y in the equation (A) (2x + 8)kg (B) (2x – 8)kg 4y – 15 = 13? (C) (8x + 2)kg (D) (8x – 2)kg (A) 3 (B) 7 (C) 12 (D) 14 What is the value of variable z in the equation 9z + 23 = 50? (A) 3 (B) 5 (C) 7 (D) 9 www.betoppers.com

6th Class Mathematics

78

11. Eric has 5 more pens than Kevin, who has p pens. The number of pens with Eric is

HOTS Worksheet 1.

The cost of an ice-cream cone is Rs.1.50 more than the cost of a particular chocolate bar. If the chocolate bar costs Rs.y, then the ice-cream cone costs (A) Rs.1.50 y (B) Rs.(y + 1.50)

 y  (C) Rs.   1.50  2.

(D) Rs.(y – 1.50)

Ben’s weight is four times Nick’s weight. If Nick’s weight is x pounds, then Ben’s weight, in pounds, is (A) x + 4

3.

4.

5.

6.

7.

(B) 4x

(C)

x 4

(D) x – 4

Seven less than a number x when multiplied by five is equivalent to the expression (A) x – 7 × 5 (B) (x – 7) × 5 (C) 7 – x × 5 (D) (7 – x) × 5 What is the value of variable p in the equation 49 – 7p = 21? (A) 3 (B) 4 (C) 5 (D) 6 What is the value of the variable m in the equation 3m – 7 = 11? (A) 3 (B) 6 (C) 8 (D) 9 What is the value of variable n in the equation 2n + 11 = 21? (A) 5 (B) 8 (C) 12 (D) 13 Which of the following expressions correctly  represents the expression “eight times a number z”?

8.

(A) 8 × z (B) 8 + z (C) z – 8 (D) z  8 Which of the following expressions correctly  represents the expression “number n divided by seventeen”?

9.

(A) n – 17 (B) 17 – n (C) n  17 (D) 17  n Which of the following expressions correctly  represents the expression “five less than a number x”?

(A) 5 – x (B) x – 5 (C) x  5 (D) 5 × x 10. Adam is 4 inches shorter than Brad, whose height is x inches. Adam’s height, in inches, is (A) x × 4 (B) x + 4 (C) x – 4 (D) x  4

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(A) p × 5 (B) p + 5 (C) p  5 (D) p – 5 12. Mrs. Thomas divided n chocolates equally among her 2 children. The number of chocolates received by each child is equivalent to which of the following expressions? (A) 2 × n (B) n + 2 (C) n  2 (D) 2 – n 13. Adam is three years elder to his brother Dan. If Adam’s age is x years, then Dan’s age is equivalent to which of the following expressions? (A) (x + 3) years (B) (x – 3) years (C) (2x + 3) years (D) (2x – 3) years 14. The number of books in Andy’s bag equals x. If Sam has 3 more books than twice the number of books with Andy, then the number of books with Sam equals (A) 2x + 3 (B) 3x + 2 (C) 2x – 3 (D) 3x – 2 15. Gaudy purchased some pens and pencils. For every pen, he purchased 2 pencils. If the total number of pens purchased was x, then the number of pencils purchased was (A) x – 2 (B) x + 2 (C) x (D) 2x

IIT JEE Worksheet I.

Single Correct Answer Type

1.

4.

Simplify: –6m – 12 + 7m – 14 (A) –13m – 26 (B) –m – 26 (C) m – 26 (D) 13m – 26 Simplify: 11r + 13 – 9r – 15 (A) 2r + 28 (B) 2r – 2 (C) 2r – 28 (D) 2r + 2 Simplify: 20p + 6 – 23p + 9 (A)–3p+15 (B) –3p –15 (C) 3p+5 (D) 3p – 15 Which of the following are like algebraic terms?

5.

–4pq, 5xy2, 6ab, 0.7pq, 9abc, 3xy, –x2yz (A) 5xy2, 3xy, –x2yz (B) –4pq, 0.7pq (C) 6ab, 9abc (D) 5xy2, 3xy State the unknowns and coefficient of –7pq.

2. 3.

Unknowns (A) p, –q (B) –p, q (C) p, q (D) p, –q

Coefficient –7 7 –7 7

Algebra 6.

7.

8.

9.

79

Which of the following is a pair of unlike algebraic terms? (A) –pqr, 0.8qrp (B) a2 bc, –6ba 2 c (C) 1.5xzy, 3xyz (D) –kmn, 48kml Given that p = –3 and q = 2, evaluate p2 + (–pq2) – p2 q2 . (A) –33 (B) 6 (C) –15 (D) 36 2 2 2 3mx – 8n y + (–4mx ) –n2y = (A) –mx2 – 9n2 y (B) 7mx2 – 9n2 y (C) –7mx2 + 9n2 y (D) 7mx2 + 9n2 y Evaluate 7p2x – (–4qy2), given that p = –2, q = 1, x = 3 and y = –4.

IV. Integer Type 18. A football team lost 5 yards and then gained 9. What is the team’s progress? 19. Use distributive property to solve the problem below: Maria bought 1 notebooks and 3 pens costing 2 dollars each. How much did Maria pay? 20. A customer pays 5 dollars for a coffee maker after a discount of 2 dollars What is the original price of the coffee maker? 21. Half a number plus 5 is 9. What is the number?

V. (A) –4 (B) 84 (C) 20 (D) 148 10. A number is divided by three and multiplied by the square of a second number. The product is then 22. divided by three. Write the algebraic term for the given statements using p as the first number and q as the second number. (A) 9pq2

pq 2 pq 2 (B) (C) 3 9

(D) 3pq2

Matrix Matching (Match the following) a b  5 ;  5 ; then 2 3 Column – I (A) ab If

a b (C) a2 + b2 (B)

II. Multi Correct Answer Type

i) Area of the rectangle = (length x width) ii) Perimeter of the rectangle = 2 (length + width) Use the above information to solve the following 15. The area of a rectangle is 24 cm2. The width is two less than the length. Find the length and width of rectangle ? 16. The Perimeter of rectangle is 20 cm. If the length of the rectangle is 6 cm .Find the width of the rectangle ? 17. If the length and width of the rectangle is give as 10 cm and 6 cm respectively. Then find the area and perimeter of the rectangle?

3 2 (r) 6 (q)

1

11. Which of the following are like algebraic terms? (D)  1  1  2 2 2 a b (A) 5y x, 3xy , –4y x (B) –4pq, 0.7pq 2 (C) 6ab, 9ba D) 5xy , 3xy 12. Which of the following is a pair of unlike algebraic terms? 23. If l = 2; m = 4 2 3 (A) –pqr, 0.8qrp (B) a bc, –6ba c Column – I (C) 1.5x2zy, 3xyz (D) –kmn, 48kml 13. Which of the following are monomials? 1 (A) 3 2 (A) 2x (B) 5x + 4 (C) 5y (D) –9x + 5y + 4 m 14. Which of the following are binomials? m (B) (A) 2x3 (B) 5x+4 (C) 5y3+5y (D) –9x2+5y+4 1

III. Paragraph Type

Column – II (p) 50

(C) lm (D) l2 m

(s)

2 3

(t) 325 (u) 150

Column – II (p) 2 (q) 16

1 2 (s) 8 (t) 32 (u) 1 (r)

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80

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6th Class Mathematics

Geometry

Learning Outcomes 

Lines and angles



Circles



Triangles



Symmetry, Reflection and Rotation



Quadrilaterals



Three dimensional figures

1. Lines and Angles Babilonians used geometry during the period 3000 BC to 2000 BC. They found formulae for areas and perimeters of rectangle and square. To mark the boundaries of the fields that were affected by the floods due to the over flow of river Nile and to compute their areas, geometry came to help of Egyptians. From ancient Vedic times geometry got special importance in our country. Geometry was developed as a science along with astronomy in our country. ‘Kalpa’ is a part of vedas. Later on, geometry was developed as ‘sulbha sutras’. Great people like Boudhayana, Apasthambha, Katyana etc., became famous authors of such ‘sulbha sutras’.

Basic Geometrical Concepts Axioms: The basic facts which are taken for granted without proof are called axioms. Eg: • Halves of equals are equal • The whole is greater than each of its parts • A line contains infinitely many points Statements: A sentence which can be judged to be true or false is called a statement. Eg: • The sum of the angles of a triangle is 1800 is a true statement • x + 8 > 12 is a sentence but not a statement Theorems: A statement that requires a proof is called a theorem. Establishing the truth of a theorem is known as proving the theorem. Eg: The sum of the angles of a triangle is 180°. Corollary: A statement, whose truth can easily be deducted from a theorem, is called its corollary.

Fundamental Geometrical Terms Point: A point is a mark of position. A fine dot represents a point. We denote a point by a capital letter A, B, P, Q etc. A point has no length, breadth or thickness.

Chapter – 5

By the end of this chapter, you will be understand

Line Segment: The straight path between two points A and B is the line segment AB, represented as AB . The points A and B are called its end points. A line segment has a definite length. A

B

The distance between two points A and B gives the length of the line segment AB . Ray: A line segment AB when extended indefinitely  in one direction is the ray AB . It has one end point A. A

B

A ray has no definite length. A ray cannot be drawn, it can simply be represented on the plane of a paper. Line: A line segment AB when extended  indefinitely in both the directions is called line AB . A

B

A line has no end points. A line has no definite length. A line cannot be drawn, it can simply be represented on the plane of a paper.

Incidence Axioms on Lines

• • •

A line contains infinitely many points An infinite number of lines can be drawn to pass through a given point One and only one line can be drawn to pass through two given points A and B. A

B

6th Class Mathematics

82 Collinear Points: Three or more points are said to be collinear, if there is a line which contains them all. P

Q

OA and OB are called the arms of the angle and O is called as its vertex. Measure of an Angle: The amount of turning from OA to OB is called the measure of  AOB is written as  AOB. An angle is measured in degrees, denoted by .

R

In the above figure; P, Q, R are collinear points. Plane: A flat surface extended endlessly in all the four directions is called a plane. The surface of a smooth wall, the surface of the top of the table, the surface of a smooth blackboard, the surface of a sheet of paper etc. are close examples of a plane. These surfaces are limited in extent but the geometrical plane extends endlessly in all directions. Two lines lying in a plane either intersect exactly at one point or are parallel.

O

A A complete rotation around a point makes an angle of 360°. 1° = 60 minutes, written as 60 . 1 = 60 seconds, written as 60 . We use a protractor to measure an angle.

Intersecting Lines: Two lines having a common point are called intersecting lines. The point common to two given lines is called their point of intersection. In the figure, the lines AB and CD intersect at a point O. C

Kinds of Angles • •

B O

A

•

•

D

Concurrent Lines: Three or more lines in a plane are said to be concurrent, if all of them intersect at the same point. In the figure below, the lines l, m, n all intersect at the same point P; so they are concurrent.

•

•

Right Angle: An angle whose measure is 90° is called a right angle. Acute Angle: An angle whose measure is more than 0° but less than 90°, is called an acute angle. Obtuse Angle: An angle whose measure is more than 90° but less than 180° is called an obtuse angle. Straight Angle: An angle whose measure is 180° is called a straight angle. Reflex Angle: An angle whose measure is more than 180° but less than 360° is called a reflex angle. Complete Angle: An angle whose measure is 360° is called a complete Angle.

n

A p

B

m

1

Parallel Lines: Two lines l and m in a plane are said to be parallel, if they have no point in common and is written as l || m. The distance between two parallel lines always remains the same. m

B O

O

Acute angle

Right angle A

l

Angles: Two rays OA and OB having a common end point O form angle AOB, written as AOB. B

O

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O

A

B Obtuse angle

A

Geometry

83 B

O B

O

A

Straight angle

120 A Reflex angle

O

50

B

A B Complete angle

• Equal Angles: Two angles are said to be equal, if they have the same measure. • Bisector of an Angle: A ray OC is called the bisector of  AOB, if m  AOC= m  BOC.

A

O

C

Important Results: Linear Pair Axiom: If a ray stands on a straight line, then the sum of the adjacent angles so formed is 1800 . Thus, AOC + COB = 1800 where AOB is a straight line. C

120

B

A

C

O

A

• Complementary Angles: Two angles are said to be complementary, if the sum of their measures is 90°. Two complementary angles are called the complement of each other. Thus, Complement of an angle of 36° = An angle of (90° – 36°) = 54°. • Supplementary Angles: Two angles are said to be supplementary, if the sum of their measures is 180°. Two supplementary angles are called the supplement of each other. Thus, supplement of an angle of 42° = An angle of (180° – 42°) = 138°. Supplement of an angle of 115° = an angle of (180° – 115°) = 65°. • Adjacent Angles: Two angles are said to be adjacent angles, if they have a common vertex and a common arm such that the other arms of the two angles are on either side of their common arm. In the adjoining figure,  AOB and  BOC are adjacent angles.

60 O

B

Converse of a Linear Pair Axiom: If two adjacent angles are supplementary, then the noncommon arms of two angles are in a straight line. Thus if two adjacent angles  AOC and  BOC with common arm OB are such that  AOC +  BOC = 180° then OA and OB are in the same straight line i.e. AOB is a straight line. C

A

O

B

Angles at a Point: The sum of all the angles at a point is 360°. In the given figure, we have  1 +  2 +  3 +  4 = 360°.

2 O 3

1 4

C B

O

A

Linear Pair: If the sum of two adjacent angles is 180°, they are said to form a linear pair. In the adjoining figure,  AOB +  COB = . So AOB and BOC together form a linear pair.

Vertically Opposite Angles: If two straight lines AB and CD intersect at a point O, then AOC and BOD form one pair of vertically opposite angles and the angles AOD and BOC form another pair of vertically opposite angles. When two lines intersect each other, then vertically opposite angles are always equal. In the given figure, we have: AOC = BOD and BOC = AOD. www.betoppers.com

6th Class Mathematics

84 A

• If two straight lines are intersected by transversal such that a pair of corresponding angles are equal, then the two lines are parallel. • If two straight lines are intersected by a transversal such that a pair of alternate angles are equal, then the two lines are parallel. • If two straight lines are intersected by a transversal such that a pair of consecutive interior angles is supplementary, then the two lines are parallel. Basic Axiom of Parallel Lines: (Euclidian postulate) • There exists one and only one line which is parallel to a given line from a given point. • If two lines are parallel to the same line then the lines are parallel to each other.

D

O C

B

Parallel Lines: Two straight lines lying in the same plane are said to be parallel if they do not intersect no matter how long they are produced on either side. In the given figure, AB and CD are parallel lines and we write AB || CD. A

B

C

D

Transversal: A straight line that intersects two or more lines is called a transversal. Let two lines AB and CD are cut by a transversal l, then the following angles are formed.

Formative Worksheet

l C

A

1

2 3 6 7

4

A

B

5 8

R

B

• Pairs of Corresponding Angles (abbreviated as corres  s): (  1,  5), (  2,  6), (  4,  8), (  3,  7) • Pairs of Alternate Interior Angles (abbreviated as Alt. Int.  s): (  3,  5) and (  4,  6) • Pairs of Alternate Exterior Angles: (  2,  8) and (  1,  7) • Pairs of Consecutive Interior Angles (abbreviated as co. Int. s): (  4,  5) and (  3,  6) Properties of Angles Associated with Parallel Lines: If two parallel straight lines are intersected by a transversal, then • Corresponding angles are equal.  1 =  5,  2 =  6,  4 =  8 and  3 =  7 • Alternate interior angles are equal.  3 =  5 and 4 = 6 • Alternate exterior angles are equal.  2 =  8 and 1 = 7 • Co-interior angles (consecutive interiors) are supplementary.  4 +  5 = 180° and  3 +  6 =180°. Conditions of Parallelism: The converse of the above results is also true.

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P

D

Q U

1. D

C T S

2.

3.

What is the number of pairs of parallel line segments in the given figure? (A) Four (B) Five (C) Six (D) Seven The pair of sides of a triangle is an example of (A) Parallel lines (B) Intersecting rays (C) Parallel line segments (D) Intersecting line segments A ray AB with end point A is symbolically represented as    (A) AB (B) AB (C) AB (D) AB

4. A

5.

B

What does the given figure represent? (A) Ray (B) Line (C) Angle (D) Line segment How many line segments can be drawn by using three different points which are not in a straight line? (A) 2 (B) 3 (C) 4 (D) 5

Geometry

85

6.

I.

I

II

III

Point E is the vertex of the angles  AEX and  BEX. II. FC is an arm of  EFD. III.  CFX and  XFD have one arm in common. IV.  CFE and  BEF have a common vertex. Among the given statements, (A) only I is incorrect (B) only IV is incorrect (C) II and III are incorrect (D) II and IV are incorrect

IV

Which two of the given figures show pairs of parallel lines? (A) II & III

(B) III & IV

(C) I & II

(D) I & IV m

7.

l

10.

n

r

q p

s

t l m n

What are the respective number of pairs of intersecting lines and parallel lines in the given figure? (A) 3 & 1 (B) 3 & 2 (C) 6 & 1 (D) 6 & 2

p q

8. What are the respective numbers of pairs of parallel lines and pairs of intersecting lines in the given figure?

9.

(A) 9 & 12 (B) 9 & 15 (C) 13 & 15 (D)13 & 18 In which of the following figures does the shaded 11. Which of the following statements is correct? portion correctly show the interior of the given (A) A line can be extended in one direction only. curve? (B) A line segment has no definite length. (A) (B) (C) A ray can be extended in both the directions. (D) The distance between two parallel lines is constant. 12. The points A, B, C, and D are collinear (the points (C) (D) lying on a straight line) such that B is the mid-point of AC and C is the mid-point of AD. If AB = 3 cm, then what is the length of BD? (A) 6 cm (B) 9 cm (C) 12 cm (D) 14 cm With respect to the given figure, four statements 13. At which of the following times is the angle between the minute hand and hour hand of a clock acute? are given as follows. (A) 1 o’clock (B) 3 o’clock A E B (C) 4 o’clock (D) 6 o’clock 14. At which of the following times is the angle between the minute hand and hour hand of a clock a straight C F D angle? (A) 1 o’clock (B) 3 o’clock (C) 6 o’clock (D) 8 o’clock X www.betoppers.com

6th Class Mathematics

86 15. What type of angle is made by minute hand and hour hand when time is 5 a.m.? (A) An acute angle (B) An obtuse angle (C) A right angle (D) A straight angle 16. A tower OA is such that AB = BC = CD = DE = EF = FG = GO and the length of AB = 8 feet.

4. A P

D

C

A

O

B Q

B C

5.

D E

Which of the following points lies in the interior region of  AOB? (A) C (B) O (C) P (D) Q What is the product of number of vertices and sum of number of sides and angles of a triangle? (A) 9 (B) 18 (C) 22 (D) 36

6. r

s

F G O

What is the height of AD? (A) 16 feet

(B) 24 feet

(C) 32 feet

q m

l

(D) 40 feet

p n

How many pairs of parallel lines are there in the given figure? (A) 3 (B) 4 (C) 5 (D) 6

Conceptive Worksheet 7. 1.

2. 3.

P

Q

R

S

How many different line segments can be named in the given figure? (A) 3 (B) 4 (C) 6 (D) 8 How many vertices does a triangle have? (A) Two (B) Three (C) Four (D) Five Which of the following geometric figures represents the line segment PQ? (A) P

Q

(B) P

(C) (D)

Q

P P

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Q Q

A R

Q

L N

P

B M

Y

X

Which of the following statements is correct with respect to the given figure? (A) Points M and X lie on the curve. (B) Points P and R lie in the exterior of the given curve. (C) Points L, M, and N lie in the interior of the given curve. (D) Points A, B, X, and Y lie in the exterior of the given curve.

Geometry 8.

87

The given figure represents a polygon.

F P

N 

C

O 

L

C.

A

 K

Which of the following sets of points lie inside the given polygon? (A) L, P, H (B) L, P, C (C) O, N, C, F (D) P, L, H, A, K Six points A, B, C, D, E, and F lie on a straight line in order such that B is the mid-point of AC , E is the mid-point of CF , and D is the mid-point of BE . If BD = 4 cm, then what is the length of AF ?

(A) 10 cm (B) 12 cm (C) 14 cm (D) 16 cm 10. Rahul is facing towards the East direction. If he

1 changes direction by turning 1 of a straight angle 4 in the anti clockwise direction, then which direction will he be facing? (A) South West (B) North West (C) West (D) North 11. The angle between the minute hand and the hour hand of a clock at 6 p.m. is equal to how many times a right angle? (A) 1

symbol . ABC has: • Three vertices, namely A, B and C • Three sides, namely AB, BC and CA • Three angles, namely, A, B and

H  A

9.

2. Triangle A closed plane figure bounded by three line segments is called a Triangle. We denote a triangle by the

G 

S 

(B) 1

1 2

(C) 2

(D) 2

1 2

B

C

The three sides and the three angles are known as elements or parts of the Triangle.

Kinds of Triangles Classification of Triangles according to Sides: Scalene Triangle: A triangle, in which all sides are of different lengths, is called a scalene triangle. In the following figure,  PQR is a scalene triangle P

Q

R

since PQ QR PR Isosceles Triangle: In an isosceles triangle, the angles opposite to the equal sides are equal. A triangle having two sides equal is called an isosceles triangle. A

12.

The angle highlighted in the given figure is (A) A right angle (B) An acute angle (C) An obtuse angle(D) A straight angle 13. The hour hand of a clock is at 4. How many right angles will the hour hand of the clock revolve in the clockwise direction to be at 10? (A) 1 (B) 2 (C) 3 (D) 4 14. Amrita is facing in the West direction. If she turns

1 1 of a revolution anti clockwise, then in which 4 direction will she face? (A) East (B) West (C) North (D) South

B

C

In  ABC; AB = AC   B =  C. Equilateral Triangle: A triangle which is having equal sides is called an equilateral triangle. The measure of each angle of an equilateral triangle is 600. A 60°

60° B

60° C

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6th Class Mathematics

88 Classification of Triangles according to Angles: Acute-Angled Triangle: A triangle, in which every angle measures more than 0° but less than 900 is called an acute-angled triangle. In the below figure PQR, we have: P = 70°, Q = 45° and R = 65°. Thus, each angle of PQR is acute.

All the three medians of a triangle are concurrent i.e. they intersect at a point Centroid: The point of intersection (concurrence) of the three medians of a triangle is called its centroid. In the below figure, the three medians AD, BE and CF intersect at the point G.

P

A

700

450

F

650

Q

E

R

Right-Angled Triangle: A triangle in which one of the angles measures 900 is called a right angled triangle or simply a right triangle. In a right-angled triangle the side opposite to the right angle is called its hypotenuse and the remaining two sides are called its legs. A

900 C

B

 ABC is a right triangle and  B = 90°, AC is the hypotenuse and AB and BC are its legs.

G

B

D

G is the centroid of

C

ABC

Altitude: The length of perpendicular from a vertex to the opposite side of a triangle is called its altitude and the side on which the perpendicular is being drawn, is called its base. In the below figure, AL BC. So BC is the base and AL is the corresponding altitude of the triangle. A triangle has three altitudes. All the three altitudes of a triangle are concurrent i.e., they intersect at a point. A

Obtuse-Angled Triangle: A triangle in which one of the angles measures more than 90° but less than 180° is called an obtuse- angled triangle. A

B

1350 C

here,  ABC is an obtuse-angled triangle in which  C =135°. Median: A line segment joining a vertex to the midpoint of the opposite side of a triangle is called a median of the triangle. In the below figure, D is the mid-point of side BC of ABC. AD is a median of ABC. A triangle has 3 medians. A

B

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D

B

Orthocenter: The point of intersection (concurrence) of the three altitudes of a triangle is called its orthocenter. In the figure below, the three altitudes AL, BM and CN of ABC intersect at a point H. Therefore, H is the orthocenter of ABC. A N

M H

B

C

C

L

L

C

Incentre and Incircle: The point of intersection of the internal bisectors of the angles of a triangle is called its incentre. In the below figure, the bisectors of the internal angles of a ABC meet at a point

Geometry

89

I. So, ‘I’ is the incentre of ABC. The incentre of a triangle is the center of a circle which touches all the sides of the triangle and this circle is called the incircle of the triangle. If ID BC, then ID is called the radius of the incircle. A

• If 1 side of a triangle is produced the exterior angle formed is equal to sum of interior opposite angles. • In a triangle the exterior angle is always, greater than its interior opposite angle. The sum of exterior angles of a triangle is 360°.

Formative Worksheet

17. Which of the following statements is correct? (A) BC is the side opposite to  ACB in I  ABC. (B) R is the vertex opposite to side TR in  RST.. B C D (C)  EGF is the angle opposite to side EF in Circumcentre and Circumcircle: The point of  GEF.. intersection of the perpendicular bisectors of the (D) MN is the side opposite to the vertex M in sides of a triangle is called its circumcentre. In the  MNO. below figure, the perpendicular bisectors of the sides 18. Which of the following figures is an obtuse-angled triangle? AB, BC and CA of ABC intersect at a point O. So O is the Circumcenter of the triangle. The A A circumcentre of a triangle is the centre of a circle which passes through the vertices of the triangle (A) C (B) and this circle is called the circumcircle of the B B C triangle. Clearly, OA = OB = OC = radius of A circumcircle. A

(C)

(D) B

C

B

C

19. 600

Exterior and Interior Opposite Angles of a Triangle: Let ABC be a triangle one of whose sides BC is produced to D; then  ACD is called an exterior angle and the angles  A and  B are called its interior opposite angles.

Properties of Triangles • Angle Sum Property: the sum of the angles of a triangle is180°. • If two sides of a triangle are equal in length, then the angles opposite to them are of equal measures. • The sum of any two sides of a triangle is always greater than the third side. • If 2 sides of a triangle are of an unequal length, then the greater side has the greater angle opposite to it.

5cm

3cm

4cm

4cm 600

600 6cm I

6cm III

II

4 cm 1200

250

450 900

350

6 cm 8 cm

IV

450 V

VI

In the given set of triangles, how many triangles are scalene? (A) 1 (B) 2 (C) 3 (D) 4

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6th Class Mathematics

90 20. Row

Measure of triangle

(a) (b) (c)

Row

ΔABC with AB = BC = CA

I

ΔPQR with P = 100°, Q = C = 40°

II

ΔWXY with W = 40,

III

Identification Acute triangle

A

23. 5cm

Scaleneright angled

B

5cm

6cm E

C

5cm

10cm 8cm

P Obtuse-angled isosceles

IV

ΔUVW with U = 90°, UV = 5 cm, VW = 13 cm, WU = 12 cm

F

Equilateral

X = 60°, Y = 80° (d)

D

The rows in the given table can be correctly matched as (A) (a)  I, (b)  IV, (c)  III, (d)  II (B) (a)  I, (b)  III, (c)  IV, (d)  II (C) (a)  III, (b)  IV, (c)  I, (d)  II (D) (a)  III, (b)  I, (c)  IV, (d)  II

S

5cm

13cm Q

6cm R

12cm

6cm

T

8cm

U

Which of the given triangles is isosceles? (A) ABC (B) DEF (C) PQR (D) STU 24. C

500

O

8cm

5cm

21. 700

What is the possible measure of the given angle  COD? (A) 80° (B) 70° (C) 50° (D) 20°

55 0

9cm

The given two triangles are respectively (A) obtuse and scalene (B) right and equilateral (C) scalene ad acute (D) isosceles and right 22. An equilateral triangle and obtuse triangle are respectively shown in which of the following pairs of triangles? (A)

D

25. P

850 O

3 cm

3 cm 700

450 3 cm

Conceptive Worksheet

(B) 4cm

4cm

200 1300

300

4cm

(C)

400 4 cm 1000

5 cm

400 6 cm

(D) 450 5 cm

900

Q

What is the possible measure of the given  POQ? (A) 80° (B) 50° (C) 45° (D) 35°

450

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5 cm

5 cm

15. How can the angle of measure 125° be classified? (A) Right angle (B) Acute angle (C) Obtuse angle (D) Straight angle 16. The angles of an equilateral triangle are (A) Acute (B) Right (C) Obtuse (D) Straight 17. Which of the following statements is incorrect? (A) In scalene triangle, all the angles are of different measures. (B) In scalene triangle, all the sides are of the same measure. (C) An isosceles triangle has exactly two equal angles. (D) An isosceles triangle has exactly two equal sides.

Geometry

91 21.

18. 4cm

4cm 5cm

6cm

3cm

(i)

(ii)

(iii)

O

6cm

B

What is the possible measure of the given angle  AOB? (A) 60° (B) 50° (C) 45° (D) 15°

5cm 500

450

A

4cm

5cm

6cm

6cm

C

700

450 (iv)

A

600

(v)

(vi)

22.

How many isosceles triangles are shown in the given set of figures? (A) 1 (B) 2 (C) 3 (D) 4 19. An acute tr iangle and a right-triangle are respectively shown in which of the following pairs of triangles?

 E

900

80 0

 O

P

45 0

 Q

O

(B) 950

900

450

 Q

O

A

450

23.

40 0

 F

With respect to the given figures, which of the following angles is a right angle? (A) AOB (B) COD (C) EOF (D) POQ

78 0 220

 D

O P

450

(A)

 B

O

 B

O

450

E

600

(C)

 C

600 900

600

300

600

35 0

(D)

1100

350

30 0

A

20.

D

700

600 700

400 B

 D

 F

O

With respect to the given figures, which of the following angles is a straight angle? (A) EOF (B) POQ (C) COD (D) AOB

3. Quadrilaterals

600 900

 O

30 0

90 0 C E

A closed figure bounded by four line segments is called a quadrilateral. A quadrilateral ABCD has: • Four vertices, namely A, B, C and D • Four sides, namely AB, BC, CD and DA • Four angles, namely  A,  B,  C and  D • Two diagonals, namely AC and BD

F D

P

C

S 600

300 120 0 Q

300

600 R

T

600 U

A

B

Which of the given triangles is a right-triangle? (A)  ABC (B)  DEF (C)  PQR (D)  STU www.betoppers.com

6th Class Mathematics

92 Adjacent Sides: Two sides of a quadrilateral having a common end point are called its adjacent or consecutive sides. In the given quadrilateral, (AB, BC), (BC, CD), (CD, DA) and (BA, AD) are four pairs of its adjacent sides. Opposite Sides: Two sides of a quadrilateral having no common end point are called its opposite sides. In the given quadrilateral (AB, CD) and (BC, AD) are two pairs of its opposite sides. Adjacent Angles: Two angles of a quadrilateral having a common arm are called its adjacent or consecutive angles. In the given quadrilateral (  A,  B), (  B,  C), (  C,  D) and (  D,  A) are four pairs of adjacent angles. Opposite Angles: Two angles of a quadrilateral not having a common arm are called its opposite angles. In the given quadrilateral (  A,  C) and (  B,  D) are two pairs of its opposite angles. • The sum of all the angles of a quadrilateral is 360°. • The sum of exterior angles of a quadrilateral is 360°. Parallelogram: A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. D

C

In a Rectangle: • Opposite sides are equal • Each angle measure is 90° • Diagonals are equal • Diagonals bisect each other Square: A Rectangle having all sides equal is called a square. If two adjacent sides of a Rectangle are equal, then it is called a Square D

C 900

900

900

900

A

B

In a Square: • All sides are equal • Each angle measures 900 • Diagonals are equal • Diagonals bisect each other • Diagonals intersect at right angles Rhombus: A parallelogram having all sides equal is called a Rhombus D

C

. A

B

In a Rhombus: • Opposite sides are parallel • All sides are equal • Diagonals bisect each other at right angles Trapezium: A Quadrilateral in which two opposite sides are parallel is called a trapezium. In diagonal divides proportionally D

A

In a Parallelogram: • Opposite sides are equal • Opposite angles are equal • Each diagonal bisects the parallelogram • If pair of opposite sides of a Quadrilateral are equal and parallel, it is a parallelogram Rectangle: A parallelogram each of whose angle measures 90° is called a Rectangle. D

C

B

A

B

If the non parallel sides of a trapezium are equal. It is known as isosceles-trapezium Kite: A Quadrilateral in which two pairs of adjacent sides are equal is known as a kite. The longer diagonal bisects the shorter diagonal.

C

A

B

A

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D

B C

Geometry

93

Relation between Quadrilaterals

Quadrilateral

Two pairs of adjacent sides

Kite or Deltoid

One pair of opposite sides are equal Non parallel sides Trapezium

Isosceles Trapezium

are equal

Opposite sides are parallel Parallelogram

all sides are equal

Opposite sides are equal and each angle is 90

Rhombus

Rectangle

Square

Polygons: A closed plane figure bounded by three or more line segments is called a polygon. The line segments forming a polygon are called its sides. The point of intersection of two consecutive sides of a polygon is called a vertex. Vertex

Side

Side

Vertex Side

Vertex

A polygon of n-sides is called n-gon. Thus, a polygon of 15 sides is a 15-gon Diagonal of a Polygon: A line segment joining any two non-consecutive vertices of a polygon is called its diagonal. Thus, in the below figure ABCDE is a polygon and each of the line segments AC, AD, BD, BE and CE is a diagonal of the polygon.

Vertex Side

E

D

Side Side

Vertex

Vertex

A

C

The number of vertices of a polygon is equal to the number of its sides. B

Triangle

Quadrilateral

Pentagon

Interior and Exterior Angle of a Polygon: An angle formed by two consecutive sides of a polygon is called an interior angle or simply an angle of the

Hexagon

Heptagon

Octagon

polygon. www.betoppers.com

6th Class Mathematics

94 Name of Polygon

Number of sides

Number of Triangles formed

Angle sum of polygon

Triangle

3

1

1 × 180° = 180°

Quadrilateral

4

2

2 × 180° = 360°

Pentagon

5

3

3 × 180° = 540°

Hexagon

6

4

4 × 180° = 720°

Heptagon

7

5

5 × 180° = 900°

Octagon

8

6

6 × 180° = 1080°

Nonagon

9

7

7 × 180° = 1260°

Decagon

10

8

8 × 180° = 1440°

Undecagon

11

9

9 × 180° = 1620°

Dodecagon

12

10

10 × 180° = 1800°

From this table, we can see that the number of triangles formed in a polygon is two less than the number of sides. So for a polygon with n sides, the number of triangles formed is n – 2. The angle sum of a polygon with n sides is (n – 2) × 180°. Therefore, S = (n – 2) 1800. In a regular polygon with n sides each interior angle is equal to

Eg: Bricks, Eraser. Number of surfaces = 6 Number of vertices = 8 Number of edges = 12 Cube: A Solid formed by the enclosure of six square plane surfaces is called a ‘cube’. Cube has – six faces, eight vertices, twelve edges. G

 n  2   180

n When each side of a polygon is produced (extended) in a clockwise or anti clockwise direction exterior angles are formed as shown. The sum of the exterior angles of any polygon is 360°. c0

H

F

E C

D A

B

Cylinder: The base and top are circular in shape. The remaining surface is called curved surface.

b0

d0 h

a0 e0

4. Three Dimensional Figures Solid State Figures: Objects with length, breadth and height/thickness are called solids. Cuboid: The solid having the four lateral sides with base and top are in rectangular shape is called ‘cuboid’. The solid formed by six rectangular surfaces is called cuboid E

Cone: The base is a circle. Its radius is denoted by r. AC is called slant height and it is denoted by ‘s’. AO is called vertical height and it is denoted by ‘h’. A is called its vertex. A

F

H H C

D

B A

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S

G

E

B

O

C

Geometry

95

Sphere: Ball, Marbles and balls in the cycle bearings are some of examples of Sphere. In the figure below,

Triangular Pyramid: If a pyramid has triangular base, then it is called a ‘triangular pyramid’. It has three triangular lateral surfaces and one triangular base. All together the triangular pyramid has four triangular faces.

‘O’ is called its ‘centre’. AO is called its radius. Its surface is called curved surface.

A O

A

r

h C E D

B

Prism: Triangular Prism. The adjoining figure shows a ‘triangular prism’. It is so called according to the shape of its base (its base is a triangle). It has five surfaces and six vertices.

The adjoining figure shows a ‘triangular pyramid’. It’s base is a triangle BCD. Lateral height of each triangle (lateral surfaces) is called the slant height which is denoted as s and the point of meeting all

F

the lateral sides. ‘A’ is called ‘apex or vertex’. AE = h is called height.

D

E

Formative Worksheet 26.

C A

B

Pyramid: You might have seen the photographs of world famous Pyramids of Egypt. Their base is in square shape and their lateral faces are in triangular How many quadrilaterals can be observed in the shape, such that all the four surfaces meet at one given figure? point. (A) 5 (B) 6 (C) 8 (D) 9 The below figure shows a ‘Pyramid’ with square 27. How many pairs of adjacent sides are there in an ABCD as its base. octagon? (A) 5 (B) 6 (C) 7 (D) 8 a, shows the side of its AB  BC  CD  DA  base. E

28.

P

Q

Z

Y

W

X

S S

D h

a

H

a

T

B

A

C

O A

R

a a

B

EO = h is called its vertical height. EC = s is called its slant height or lateral height. The four triangular shaped surfaces and square base all together there are five surfaces.

M

A

D

C

Which of the following statements is correct with respect to the given figure? (A) PQ and RS are the adjacent sides of quadrilateral PQRS. (B) AB and BC are the opposite sides of quadrilateral ABCD. www.betoppers.com

6th Class Mathematics

96 (C) In quadrilateral WXYZ,  XWZ, and  XYZ form a pair of opposite angles. (D) In quadrilateral MATH,  MAT and  MHT form a pair of adjacent angles. 29. With respect to the given figure, some statements are given as I. AC is the side opposite to  ABC as well as  ADC. II. CD is the side opposite to  ACD. III. CD is the side opposite to  CAB. IV. Point C is the vertex opposite to side AD in

 ADC. V.  CAD is the angle opposite to side BC. VI. Point C is the vertex opposite to side AB in  ABC. A

B

D

31. Which of the following solids has a circular face? (A) Cone (B) Sphere (C) Square pyramid (D) Triangular prism 32. Which of the following three-dimensional shapes has five faces? (A) Cube (B) Square pyramid (C) Rectangular prism (D) Triangular pyramid 33.

What are the respective number of edges, faces, and vertices of the given solid? (A) 10, 6, and 6 (B) 11, 6, and 7 (C) 12, 7, and 7 (D) 13, 7, and 8 34. Which quadrilateral has always equal sides and equal diagonals? (A) Trapezium (B) Rhombus (C) Rectangle (D) Square 35. John made a scaled down model of the given figure.

C

Which of the given statements are incorrect? (A) I and II (B) III and IV (C) II, III, and V (D) II, IV, and VI 30. With respect to the given figure, some statements are given as I. Points B and C lie in the interior of  POQ as well as  UVW.. II. Points I and Z lie on the angle  POQ. III. Points F, X, K, and D lie in the exterior of  POQ as well as  UVW.. IV. Points A, E, and Y lie in the interior of  POQ. V. Points A and L lie in the exterior of  UVW.. Q P

F

A

U E

D

B I

Z

Y

C

V

W K

X

L O

Which two statements are ‘incorrect’? (A) I and II (B) II and IV (C) III and V (D) I and IV www.betoppers.com

Which of the following figures represents a possible model built by John?

(A)

(B)

(C)

(D)

36. The number of pairs of parallel lines in a triangular prism is (A) 2 (B) 4 (C) 6 (D) 8 37. In which of the following quadrilaterals are the diagonals always equal and perpendicular? (A) Parallelogram (B) Rhombus (C) Rectangle (D) Square 38. Which of the following statements is correct? (A) Every rectangle is square. (B) Every square is a rhombus. (C) Every rhombus is a rectangle. (D) Every parallelogram is a rhombus.

Geometry

97

39. Which of the following statements is ‘incorrect’? (A) Cube is a prism (B) Cube is a cuboid (C) Square is a rhombus (D) Parallelogram is a trapezium 40. The information in which alternative is correctly matched? (A) Three-dimensional figure

Attributes

Cube

Flat faces = 6 Vertices = 10 Edges = 12

(B) Three-dimensional figure

(C) Three-dimensional figure

(D)

Attributes

Sphere

Flat faces = 1 Vertices = 1 Edges = 0

Three-dimensional figure

Attributes

Square pyramid

Flat faces = 5 Vertices = 6 Edges = 9

41. The information in which alternative is correctly matched? (A)

Polygon

Polygon

Classification

Hexagon

(D)

Polygon

Classification

Heptagon

Attributes Flat faces = 6 Vertices = 8 Edges = 12

Cuboid

(C)

Classification

Triangle

42. Which of the following statements is correct? (A) If, in quadrilateral ABCD,  A =  B =  C =  D = 90°, then it is a trapezium. (B) If, in quadrilateral ABCD, AB = BC = CD = DA, then it is a rhombus. (C) If, in quadrilateral ABCD, AB||CD and BC||DA, then it is a rectangle. (D) If, in parallelogram ABCD, a pair of adjacent angles is equal, then it is a square. 43. Some statements about types of quadrilaterals are given as I. In parallelograms and rectangles, opposite sides are equal. II. In rectangles, diagonals are equal. III. In rhombuses, diagonals are always equal and perpendicular. IV. In trapeziums, exactly a pair of opposite sides is always equal. V. In squares, all the sides and angles are equal. Which two statements are ‘incorrect’? (A)I & IV (B)II & V (C)III & I (D)III & IV

Conceptive Worksheet 24. Which of the following figures is ‘not’ a hexagon?

(B)

Polygon

Classification

(A)

(B)

(C)

(D)

Quadrilateral

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6th Class Mathematics

98 25. How many diagonals can be drawn in a sevensided polygon? (A) 15 (B) 14 (C) 13 (D) 12 26. With respect to the given figure, some statements are given as I. The shaded portion represents the exterior of  ADE lying in quadrilateral ABCE. II. Points E and A lie in the exterior of quadrilateral ABCD. III. ED and DC are the adjacent sides of quadrilateral ABDE. IV.  EAB and  ECB are the opposite angles of quadrilateral ABCE. V. The pair of sides (AB and ED) and (AE and BD) are the opposite sides of quadrilateral ABDE. VI. Points A and B lie on quadrilateral ABDE as well as quadrilateral ABCE. VII.  ABC and  BCD are the adjacent angles of quadrilateral ABCD. A

B

E

D

C

Which of the given statements are ‘incorrect’? (A) I, II, V, and VII (B) II, III, IV, and VI (C) I, V, and VI (D) II and III 27. With respect to the given figure, some statements are given as G

A

28. If a quadrilateral is a rectangle, but not a square, then which of the following statements is not correct? (A) Diagonals are equal. (B) All sides are not equal. (C) Measures of all the angles are equal. (D) Diagonals are perpendicular to each other. 29. A quadrilateral is a parallelogram but not a rectangle, if (A) Diagonals are not equal (B) Opposite angles are not equal (C) Opposite sides are not parallel (D) Diagonals do not bisect each other 30. In which of the following quadrilaterals are opposite angles not equal? (A) Rhombus (B) Rectangle (C) Trapezium (D) Parallelogram 31. What is the difference between the numbers of edges of a cube and a triangular pyramid? (A) 1 (B) 2 (C) 5 (D) 6 32. The number of which of the following polygons is two more than the number of sides of a triangle? (A) Quadrilateral (B) Pentagon (C) Octagon (D) Hexagon 33. Maria’s teacher asks her to collect water in a cylindrical vessel in order to measure its capacity. The vessel looks like (A) (B)

D

S Q X

R

(C)

P

(D)

Y

W T C B Z

I. Points Q and R lie in the interior of  ABC as well as  ABD. II. Points P and W lie on  ABC as well as  ABD. III. Points S, T, and Y lie in the exterior of  ABD. IV. Points T, Z, and S lie in the exterior of  ABC. V. Points X, G, and Z lie in the exterior of  ABC as well as  ABD. VI. Points Q, S, and R lie in the interior of  ABD. Among the given statements, the correct ones are (A) I, II and VI (B) I and VI (C) III, IV and V (D) V and VI www.betoppers.com

34. The number of pairs of parallel lines in a triangular pyramid is (A) 2 (B) 3 (C) 4 (D) 5 35. Which of the following three-dimensional figures has 5 faces and 8 edges? (A) Cube (B) Cuboid (C) Square pyramid (D) Triangular pyramid

Geometry

99

36. The information given in which alternative is incorrectly matched? (A) Three Number of Number of dimensional faces edges shape Rectangle pyramid

5

8

(B)

Object

Classification Pyramid

(C)

Object

Classification Cone

Three (B) dimensional shape

Number of Number of faces edges

Triangle Prism

(C)

Three dimensional shape

5

(D)

Three dimensional shape

Object

Classification

Number of Number of faces edges

Cuboid

(D)

9

6

12

Number of Number of faces edges

Triangular pyramid

5

9

37. Which of the following statements is ‘incorrect’? (A) A hexagon has 9 diagonals. (B) A triangular prism has 5 faces, 9 edges, and 7 vertices. (C) The diagonals of a square are equal and perpendicular to each other. (D) If the length of the diameter of a circle is 12cm, then its radius is 6 cm. 38. Which of the following statements is ‘incorrect’? (A) Cube is a prism. (B) Cube is a cuboid. (C) Square is a rhombus. (D) Parallelogram is a trapezium. 39. Which of the following three-dimensional figures has 8 edges and 5 faces? (A) Square pyramid (B) Triangular pyramid (C) Rectangular prism (D) Triangular prism 40. The information in which alternative is ‘incorrectly’ matched? (A) Object Classification Cube

Cube

41. Which of the following polygons is not an octagon? (A)

(B)

(C)

(D)

5. Circle Geometric definition of a Circle: A circle is a simple closed curve consisting of all points in a plane which is at a fixed distance, say r cm, from a fixed point ‘O’ inside it. r cm O

The fixed point ‘O’ is called the centre of the circle. The constant distance r is called the radius of the circle. Radius: A line segment joining any point on the circle to its centre is called a radius of the circle. Thus, if ‘O’ is the centre of a circle and A is a point on the circle, then OA is a radius of the circle. The plural of radius is radii. By joining any point on the circle with ‘O’, we get a radius. So, we can draw an infinite number of radii of a circle. Clearly, the lengths of all the radii of a circle are equal.

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6th Class Mathematics

100

such that OP = r forms the circle. Clearly, the centre of a circle lies in its interior. P

A

O

O

r

Terms associated with a Circle Interior and Exterior of a Circle: Let us consider a circle with centre ‘O’ and radius ‘r’. Clearly, a point ‘P’ is said to lie • Inside the circle, if OP < r. • Outside the circle, if OP > r. • On the circle, if OP = r. Thus, the circle divides the plane into three parts:

Concentric Circles: The Circles with same centre and with different radii are called concentric circles. Note: Energy will propagate in concentric circles. Circular Region: The set of all points of the plane which lie either on the circle or inside the circle forms the circular region.

Exterior Interior

1.

Interior of the Circle: The part of the plane consisting of all those points whose distance from the centre of the circle is less than the radius of the circle is called the interior of the circle. In other words, the set of all points P of the plane such that OP < r forms the interior of the circle. Thus, in the following figure, the shaded region represents the interior of the given circle.

Chord: A line segment whose end points lie on a circle is called a chord. In the below figure each of the line segments PQ, RS, AB and CD is a chord of the circle with centre O. Clearly, an infinite number of chords may be drawn in a circle. Q

P R

S

A

B

O

P

C O

2.

r

Exterior of the Circle: The part of the plane consisting of all those points, whose distance from the centre of the circle is greater than the radius of the circle is called the exterior of the circle. In other words, the set of all points P of the plane such that OP > r forms the exterior of the circle. Thus, in the adjoining figure, the shaded region represents the exterior of the given circle. P

O

r

The Circle: The part of the plane consisting of all those points, whose distance from the centre of the circle is equal to the radius of the circle is the circle. In other words, the set of all points P of the plane www.betoppers.com

D

Diameter: Chord of a circle passing through its centre is called a diameter of the circle. Thus, in the following figure, AB is a diameter of a circle with centre O. Also, AB = AO + OB = 2AO [ AO = OB = radius] Therefore diameter = 2 radius

A

O

B

Note: A diameter is the longest chord of a circle. An infinite number of diameters of a circle may be drawn. Secant: A line which intersects the circle at two distinct points is called a secant. In the below figure, secant XY intersects the circle at the points M and N.

Geometry

101 A

Y P

N

O

B

Q

M X

Tangent: A line which touches a circle at one point only is called a tangent to the circle at that point. In the below figure, PTQ is a tangent to the circle at point T. The point at which the tangent touches the circle is called the point of contact.

Thus, we have: • Length of minor arc < Length of the semicircle • Length of major arc > Length of the semicircle Semicircular Region: The shaded region enclosed by semi-circle APB and the diameter AOB together with the semicircle and the diameter is called a semicircular region. P

A

O

P

T

Q

Note: One and only one tangent can be drawn to a circle at a point. Arc: Any part of a circle is called an arc. The arc of a circle is denoted by the symbol  Usually, we name an arc by three points, out of which two are the end-points of the arc and the third one lies in between them. In the adjoining figure,  is an arc of a circle with centre O. Also ACB

B

O

Circumference: The whole arc of the circle is called its circumference. Thus, the length of the circumference of a circle gives its perimeter. Angle subtended by an Arc: The angle formed by the two radii at the ends of an arc of the circle, at the centre of the circle is called the angle subtended by the arc or the central angle. In the below figure,  AOB is the angle subtended by the arc AB of a circle with centre O

 is an arc of the circle. ADB

O

C

A

A

B

O D

Major and Minor Arcs: An arc less than one-half of the whole arc of a circle is called a minor arc of the circle. An arc greater than one-half of the whole arc of a circle is called a major arc of the circle.  is the minor arc Thus, in the above figure, ACB  is the major arc. and ADB Semicircle: One-half of the whole arc of a circle is called a semicircle. Let AB be a diameter of a   as well as AQB circle with centre O. Then, APB are semicircles.

B

Segment of a Circle: The part of the circular region enclosed by an arc and the chord joining the end points of the arc together with the arc and the chord is called a segment of the circle. The segment containing the minor arc is called the minor segment, while that containing the major arc is called the major segment. Thus, the centre of the circle lies in the major segment. In the below figure, O is the centre of the circle; PRQP is the minor segment and QRSQ is the major segment. P Q

Minor segment

R

O Major segment S

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6th Class Mathematics

102 Sector of a Circle: The part of the circular region bounded by the arc and the two radii at the ends of the arc together with the arc and the radii is called a sector of the circle. The sector containing the minor arc is called the minor sector while the one containing the major arc is called the major sector. Thus, in the given circle with centre O, we have OPRQO as the minor sector and OPSQO as the major sector. S Major sector

If, in the given figure, O represents the centre of the circle, then which of the following line segments represents the radius of the circle? (A) AB (B) BE (C) CD (D) OA 48. With respect to the given circle with centre C, some statements are given below. I. The centre of the circle lies in the exterior of the triangle AYZ. II. The shaded region represents a segment of the circle. III. In the given circle, two radii are represented. IV. In the given circle, six chords are represented. X

O Minor sector

P

Q

Y

Z

C

R

Quadrant: If the two radii OP and OQ of a circle are at right angles to each other, then the sector OPRQ is called the quadrant of the circle.

O

Q R

P

Formative Worksheet 44. The length of the radius of a circle is 8 cm. If the radius is made three times, then what will be the diameter of the newly formed circle? (A) 12 cm (B) 24 cm (C) 48 cm (D) 96 cm 45. If the length of the longest chord of a circle is 12 cm, then what is the length of the radius of the circle? (A) 3 cm (B) 6 cm (C) 10 cm (D) 12 cm Y

46.

X B O

B

D A

Which of the given statements are ‘incorrect’? (A) I & II (B) II & III (C) III & IV (D) I & IV 49. With respect to the given circle with centre C, some statements are given as I. The shaded portion represents a segment of the circle. II. PQ is the circumference of the circle. III. CX and PQ are the diameters of the circle. IV. PR and PQ are the chords of the circle. R

Q C

P

X

Among the given statements, (A) Only statement II is incorrect (B) Statements II and III are incorrect (C) Statements I, II, and III are incorrect (D) Statements II, III, and IV are incorrect 50. The given figure shows a circle with centre O. T

A

Q

C

How many radii are there in the given figure? (A) 3 (B) 4 (C) 5 (D) 6

P O

D

47. C A

R

S O

E

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B

U

V

How many radii does the given circle have? (A) 3 (B) 6 (C) 7 (D) 9

Geometry

103

(A) Shaded portion I is a sector of the circle. (B) Only two radii are shown in the circle. 42. How many times the length of the radius of a circle (C) Shaded portion II is a semi-circle. is the length of the chord passing through the centre (D) AB is the diameter of the circle. of the circle? 48. The given figure shows a circle with centre O. A (A) One (B) Two (C) Three (D) Four 43. Minor segment of a circle is the region bounded by E (A) minor arc and a chord B (B) major arc and a chord O (C) minor arc and two radii (D) major arc and two radii D

Conceptive Worksheet

44.

Z O

C

How many chords does the given circle have? (A) 7 (B) 8 (C) 9 (D) 10 49. The given figure shows a circle with major segment AXB.

y

X

In the given circle with diameter XZ and radius OY, the center of the circle is represented by point (A) O (B) X (C) Y (D) Z 45. The given figure shows a circle. AB and CD are the diameters of the circle and intersect at point R.

A

D P 

A

 Q

 R

X

 S

B

C

In the given figure, the center of the circle is denoted by the point (A) P (B) Q (C) R (D) S F

46.

B

A B

O C E D

How many chords are shown in the given circle? (A) 2 (B) 3 (C) 4 (D) 5 47. The given figure shows a circle with centre O, where A, B, C, and D are points on the circle and AB is a line segment.

What type is the angle whose end points are A and B and vertex is X? (A) Acute angle (B) Obtuse angle (C) Right angle (D) Straight angle

6. Symmetry, Rotation

Reflection

and

Types of Symmetry Linear Symmetry: A figure is said to possess a linear symmetry if there exists a line which divides the figure into two parts such that if the figure is folded along this line then the part of the figure on one side of the line falls exactly over the part on the other side. We say that the figure is symmetrical about this line and the line is called the axis of symmetry or the line of symmetry. Eg: Consider the following figures: P

A

II O

A

B

I C

D

Which of the following statements is ‘incorrect’ with respect to the given figure?

Q

B

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6th Class Mathematics

104 M

Rotational Symmetry: Possesses a rotational symmetry of order 2 about the mid-point O of the segment. P

A

B O

Q

N

•

Clearly each of the figures is divided into two coincidental parts by the dotted line. Therefore, all these figures possess a linear symmetry and in each case, the dotted line is the line of symmetry. Note: Each one of these figures has only one line of symmetry. Point Symmetry: A figure is said to possess a point symmetry about a point O, called the centre of symmetry if corresponding to each point P on the figure, there exists a point P on the figure such that V lies directly opposite to the point P on the other side of the centre O. A figure that possesses a point symmetry attains its original form upon being rotated through 180  .

An angle AOB having equal arms OA and OB Linear Symmetry: One line of symmetry i.e. bisector of the angle AOB. Point Symmetry: No point of symmetry. Rotational Symmetry: No rotational symmetry. A Q

B

O P

•

A Scalene Triangle ABC Linear Symmetry: No line of symmetry. Point Symmetry: No point of symmetry. Rotational Symmetry: No rotational symmetry. A

Letters of the English alphabet have one or more lines of symmetry

B

•

C

An Isosceles Triangle ABC having AB = AC Linear Symmetry: One line of symmetry i.e. bisector PQ of the angle included between equal sides. Point Symmetry: No point of symmetry. Rotational Symmetry: No rotational symmetry. A

The following letters of English alphabet possesses a point symmetry. HINOSXZ Discussion of all the 3 types of Symmetries for various Geometrical Figures: • A Line Segment AB Linear Symmetry: One line of symmetry i.e. perpendicular bisector PQ of the line segment AB. Point Symmetry: Possesses a point symmetry the mid-point O of the line segment as the centre symmetry. www.betoppers.com

B

•

Q

C

An Equilateral Triangle ABC Linear Symmetry: Three lines of symmetry i.e. bisectors of the three interior angles. Point Symmetry: No point of symmetry. Rotational symmetry: Possesses a rotational.

Geometry

105 • A O B

•

C

A Kite ABCD in which AB = BC and CD = DA Linear Symmetry: One line of symmetry i.e. the diagonal BD. Point Symmetry: No point of symmetry. Rotational Symmetry: No rotational symmetry.

A regular Hexagon Linear Symmetry: Six lines of symmetry i.e. three diagonals passing through the centre O and three lines joining the mid-points of the opposite sides of the hexagon. Point Symmetry: Possesses a point symmetry with the centre O of the hexagon as the centre of symmetry. Rotational Symmetry: Possesses a rotational symmetry of order 6 about the centre O of the hexagon. A

B

A

B

C

F

C

O

D

•

A Semi-Circle having Diameter AB Linear Symmetry: One line of symmetry i.e. perpendicular bisector of the diameter AB. Point Symmetry: No point of symmetry Rotational Symmetry: No rotational symmetry.

E

•

P

A

D

A Circle with Centre ‘O’ Linear Symmetry: An infinite number of lines of symmetry i.e. all possible diameters of the circle. Point Symmetry: Possesses a point symmetry with its centre O as the centre of symmetry. Rotational Symmetry: Possesses a rotational symmetry of an infinite order about the centre O.

B

Q

•

A Regular Pentagon Linear Symmetry: Five lines of symmetry each being the perpendicular bisector of a side of the pentagon. Point Symmetry: No point of symmetry Rotational Symmetry: Possesses a rotational 51. symmetry of order 5 about the point O of intersection of the perpendicular bisectors of the sides of the pentagon.

Formative Worksheet

A

B

E

? Which figure correctly represents the reflected image of the given figure about the dotted line? (A) (B)

O

D

C

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6th Class Mathematics

106 (C)

(D)

56. The given figure shows a regular pentagon.

52.

C? Which figure correctly represents the reflected image of the given figure about the dotted line? (A) (B)

(C)

How many lines of symmetry does the given figure have? (A) 1 (B) 2 (C) 3 (D) 4 57.

(D)

53. Which of the following figures shows a figure and its flipped image about the dotted line? (A)

The number of lines of symmetry present in the given figure is (A) 2 (B) 4 (C) 6 (D) 8 58.

(B)

(C)

(D)

How many lines of symmetry does the given figure have? (A) 1 (B) 2 (C) 3 (D) 4 59. Which of the following figures correctly represents a figure and its reflected image about the dotted line?

54.

How many lines of symmetry does the given figure have? (A) 2 (B) 4 (C) 6 (D) 8

(A)

(B)

(C)

(D)

55.

How many lines of symmetry does the given figure have? (A) 1 (B) 2 (C) 3 (D) 4 www.betoppers.com

Geometry

107

60. Which of the following figures correctly represents a figure and its reflected image about the dotted line? (A) (B)

(C)

64. Which of the following figures correctly shows a figure and its reflected image about the dotted line?

(D)

(A)

(B)

(C)

(D)

65. Which of the following figures correctly shows a figure and its reflected image about the dotted line? 61. Which of the following figures correctly represents a figure and its reflected image about the dotted line? (A)

(B)

(A)

(B)

(C)

(D)

66. Which of the following figures correctly shows a pentagon drawn on grid paper and its line of symmetry? (C)

(D)

62. Which of the following figures show the correct lines of symmetry of the shape? (A)

(B)

(A)

(B)

(C)

(D)

Conceptive Worksheet

?

50. (C)

Which figure correctly represents the reflected image of the given figure about the dotted line?

(D)

(A)

(B)

(C)

(D)

63. A square and a trapezoid are drawn on a grid paper, as shown in the given figure.

51. The numbers of line(s) of symmetry of the square and trapezoid drawn in the given figure are respectively (A) 4 & 1 (B) 2 & 0 (C) 4 & 0 (D) 2 & 1

Which figure correctly represents the reflected image of the given figure about the dotted line? (A)

(B)

(C)

(D) www.betoppers.com

6th Class Mathematics

108 52. Which of the following figures shows a line of symmetry of the figure? (C) (A)

(C)

(D)

(B) 58. Which of the following figures correctly represents a figure and its reflected image about the dotted line?

(D)

53. Which of the following figures shows a line of symmetry of the figure?

(A)

(B)

(C)

(D)

(A)

(B)

(C)

(D)

54. How many lines of symmetry does the figure have? (A) 1 (B) 3 (C) 5 (D) 7

59. The given figure shows a shape drawn on grid paper.

55.

How many lines of symmetry does the figure have? (A) 0 (B) 1 (C) 2 (D) 3 56.

How many lines of symmetry does the figure have? (A) 1 (B) 2 (C) 3 (D) 4 57. Which of the following figures correctly represents a figure and its reflected image about the dotted line?

(A) www.betoppers.com

(B)

How many lines of symmetry does the given figure have? (A) 1 (B) 2 (C) 3 (D) 4 60. A hexagon and a trapezoid are drawn on a grid paper, as shown in the given figure.

The number of lines of symmetry of the given hexagon and trapezoid are respectively (A) 1 & 2 (B) 2 & 1 (C) 1 & 1 (D) 2 & 2

Geometry

109

61. Which of the following figures correctly shows a figure and its reflected image about the dotted line?

(A)

(C)  AOC contains the angle having vertex O and the rays OA and OC (D)  DOB contains the angle having vertex O and the rays OC and OB .

(B) 2.

X

(C)

 Z

(D) D

C

62. Which of the following figures correctly shows a figure and its reflected image about the dotted line? A

(A)

(B)

(C)

(D)

W 

B

Y

The shaded portion of which figure correctly shows  the region enclosed by  ZAW and line XY ? (A) X  Z D

C

A

B

63. The given figure shows a shape drawn on grid paper.

(B)

W  Y

X  Z D

How many lines of symmetry does the given figure have? (A) 1 (B) 2 (C) 3 (D) 4

Summative Worksheet 1.

A

C

B

(C)

W  Y

X  Z

A

C

1

O

D

D

C

A

B

(D)

W  Y

X  Z

B

Which of the following statements is incorrect with respect to the given figure? (A) The name of angle 1 is  COD (B) The vertex of  COB is point O

D

C

A

B

W  Y

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6th Class Mathematics

110 3.

8. D

A

A line segment, which passes through the center of the circle and connects two points on the circle, is known as the

O

(A) radius

B C

What are the arms of  AOC in the given figure?     (A) OA and OC (B) OB and OC

9.

(B) diameter

(C) chord (D) circumference The given figure shows a geometrical construction. A

F

E

(C) points A and C (D) points O and A A

4.

B

B

H

C

C

O

H

G

D

It can be seen that BC intersects with G

D F

E

How many diagonals containing point B or point D can be drawn in the given polygon?

5.

(A) 7

(B) 9

A

C

(C) 10

(D) 12

E

G

(A) AB, EF and CD

(B) AB, EF and GH

(C) OG, CD and GH

(D) AB, FG and CD

10. Which of the following quadrilaterals is not a parallelogram? 4cm

(A)

(B)

5 cm

B Y Z

X

D

F

W

U

V

5 cm

4 cm

S

(C)

(D)

With how many line segments is the given polygon formed? (A) 16 6.

110

T

(B) 15

(C) 14

(B)

(C)

(D)

the perpendicular bisector of AB? (A)

C

A

B

D

7.

C

(B) I

II

III

IV

Which of the given figures are not polygons? (A) I and II

(B) II and III

(C) I and III

(D) II and IV

A

B

D

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110

70

11. In which of the following figures does CD act as

(D) 13

Which of the following figures is not a polygon? (A)

70

Geometry

111 15. The angles of measure 90° and 145° are respectively (A) Right and acute (B) Acute and straight

C

(C)

(C) Right and obtuse A

(D) Obtuse and right

16. The angles of measure 36° and 105° are respectively

B

(A) Acute and obtuse (C) Right and acute

(B) Obtuse and right (D) Acute and straight

17. Which of the following rows is correctly matched?

D

C

(D)

(A) A

B

(B) D

Measure of angle

Classification

30°

Straight

Measure of angle

Classification

90°

Acute

Measure of angle

Classification

120°

Obtuse

Measure of angle

Classification

180°

Right

12. The given figure shows a quadrilateral PQRS. Its diagonals are the perpendicular bisectors of each

(C)

other. P

Q

(D)

18. Shyla stacks four triangular prisms together such that one edge is common to all the prisms as shown in the figure.

O

R

S

If PR = 10 cm and QS = 12 cm, then what is the value of SO + OR? (A) 12cm (B) 11 cm (C) 9 cm (D) 7 cm A

13.

E

B

D

C

How many acute angles are shown in the given figure? (A) 8

(B) 9

(C) 12

(D) 13

14. The marked angle in which of the following figures is a reflex angle? (A)

(B)

The resultant 3-D shape is of (A) Triangular prism (B) Triangular pyramid (C) Rectangular pyramid (D) Rectangular prism 19. A pentagonal based pyramid has (A) Five triangular faces and a pentagonal face (B) Four triangular faces and a pentagonal face (C) Three triangular faces and a pentagonal face (D) Two triangular faces and a pentagonal face 20. Which of the following figures correctly shows a figure and its reflected image about the dotted line?

(C)

(D)

(A)

(B)

(C)

(D)

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6th Class Mathematics

112 21. Three figures drawn on a grid paper are marked as I, II, and III.

HOTS Worksheet L A

C

I

P

M II

R

G S

1.

K B H

III

D F

What is the arrangement of the given figures in the increasing order of their lines of symmetry? (A) I, II, III (B) I, III, II (C) II, III, I (D) III, II, I 22. The given figure shows two shapes drawn on grid paper.

Which of the following statements is ‘incorrect’ with respect to the given figure? (A) Points A, R, C, and D lie outside the curve. (B) Points F, D, L, and M lie outside the curve. (C) Points G, S, K, and H lie inside the curve. 2.

(D) Points F, L, and B lie on the curve. The information in which alternative is correctly matched? (A)

C urve

Classification Simple curve

The number of lines of symmetry of figures I and II are respectively (A) 2 and 1 (B) 1 and 2 (C) 4 and 1 (D) 1 and 4 23. Which of the following quadrilaterals has four lines of symmetry? (A) Rectangle (B) Square (C) Trapezoid (D) Rhombus 24. Which of the following figures correctly shows the reflection of the shape with respect to the dotted line?

(B)

Curve

Classification Clos ed curve

(C)

Curve

Classification Open curve

(A)

BUS

BUS (D)

Curve

Classification

BUS

BUS

Simple curve

(B)

(C)

BUS www.betoppers.com

BUS

(D)

S UB

BUS

Geometry 3.

113 m2

m1

10. Which of the following figures correctly shows the reflected image of the shape about the dotted line?

n B

A

l1

C

D

(A)

l2

(B)

4.

5.

What are the respective numbers of pairs of parallel lines and pairs of intersecting lines in the given figure? (A) 1 and 10 (B) 2 and 10 (C) 1 and 8 (D) 2 and 8 To represent an arc AC of a circle, which of the following symbols is ‘correct’?   (B) AC (C) AC (D) AC (A) AC In the given figure, the shaded portion shows sector AOB of the circle made by the arc AB.

(C)

(D)

O

A

B

Which of the following expressions correctly represents arc AB?    (A) AC (B) AC (C) AC (D) AC 6.

11. Which of the following figures has the ‘maximum’ lines of symmetry? (A) (B)

A

(C) B

7.

(D)

C

With respect to the ABC shown in the given figure, point A lies (A) Inside the triangle 12. Two circular wheels are rolling on a horizontal road. (B) Outside the triangle (C) On the intersection of sides AB and BC of The loci of the centers will be ? the triangle 13. If one angle of a triangle is equal to the sum of the (D) On the intersection of sides AB and AC of other two, the triangle is ? the triangle Which of the following expressions ‘correctly’ 14. What value of ‘x’ will make AOB a straight line represents a ray PQ? B C    (D) PQ (A) PQ (B) PQ (C) PQ 0 2x + 30

8.

9.

A polygon with eight sides is called a (A) Square (B) Pentagon (C) Nonagon (D) Octagon Which of the following polygons is has six sides? (A) Rectangle (B) Pentagon (C) Parallelogram (D) Hexagon

2x – 500 O

A

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6th Class Mathematics

114 15. What value of x will make CD || EF if AB || CD A

B

E

F

650

X 300 D

C

16. In the given figure,  QPB = R

Q

3x 2x A

x P

22. The angles of a triangle in ascending order are x, y, z and y – x = z – y = 10°. The smallest angle is? 23. If 2 parallel lines are intersected by a transverse line, then the bisectors of the interior angles form a? 24. The number of lines of symmetry in a parallelogram is? 25. The ratio of the sides of two regular polygons is 1 : 2 and their interior angles is 3 : 4, then the number of sides in each polygon is? 26. Any cyclic parallelogram is a ? 27. In the figure below, AB = AC and  BAC = 40°. Find the sum of angle ADC and angle DAC A

B

17. The measurement of each angle of a polygon is 160°. The number of its sides is ? 18. In the given figure, AB || FD , then to A

EFD is equal

D

D

B

C

IIT JEE Worksheet 600 B

I.

E

300

Single Correct Answer Type

1. A

F

19. In the figure if BD || EF, then

CEF is

E

A

D

F C B

Which of the following statements is correct with respect to the given figure?   (A) AB and CD are intersecting rays

400 B

C

D

20. In the figure PQ || ST, then

(B) AB and CD are line segments   (C) AB and CD are parallel lines   (D) AB and CD are lines

QRS is equal to

Q

P

S

1000

T

1100

2.

Crispin draws five polygons on his notebook.

R

21. In the given figure is equal to

A

A  B  C  D  E

B

A B

C

D

E

Which two polygons are quadrilaterals? E

C

D

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(A) A & E

(B) A & D (C) B & D (D)B & C

Geometry 3.

115

If O represents the centre of the following circles,

II. Multiple Correct Answer Type

then which of the following figures correctly

7.

represents the diameter of the circle? (A)

(B)  O

 O

(C)

(D)  O

8.  O

9.

4.

10. I

II

III

IV

Which of the given figures shows a polygon? (A) I (B) II (C) III (D) IV

11.

12.

5. 13. (i)

(ii)

14.

(iii)

(iv)

Which of the given figures is a polygon? 6.

15.

(A) I (B) II (C) III (D) IV The angles of measure 180° and 75° are respectively (A) Right and acute (B) Obtuse and right 16. (C) Acute and straight (D) Straight and acute

Which of the following statements is false? (A) Through three collinear points, we can draw three lines (B) Two intersecting lines in a plane determine a point (C) Lines belonging to the same plane are called coplanar lines (D) Plane is a part of a line An angle will have: (A) No vertex (B) One vertex (C) Two arms (D) Two vertices Points A,B,C,D in this order are on a line l. Then which of the following statements is true? (A) AB and CD are two segments (B) BA and BD are two opposite rays (C) AB and BD are two opposite rays (D) CA and CD are opposite rays ______________ is an acute angle. (A) 91° (B) 100° (C) 89° (D) 1° A pair of adjacent angles will have: (A) No common arm (B) Two common arms (C) A common arm (D) A common vertex Which of the following can be sets of angles of a triangle? (A) 75°, 45°, 60° (B) 100°, 40°, 50° (C) 110°, 50°, 20° (D) 30°, 600, 90° Which of the following can be the sets of angles of an acute triangle? (A) 60°, 60°, 60° (B) 45°, 55°, 80° (C) 70°, 70°, 40° (D) 90°, 40°, 50° Which of the following statements is false? (A) If two angles of a triangle are complementary, it is a right triangle. (B) In a right triangle, the side opposite to the right angle is called the hypotenuse (C) A triangle must have three acute angles (D) A triangle can have two obtuse angles Which of the following can be the measures of the sides of a triangle? (A) 3 cm, 3 cm, 5 cm (B) 6 cm, 6 cm, 6 cm (C) 6 cm, 4 cm, 2 cm (D) 10 cm, 6 cm, 3 cm A rectangle is a: (A) Quadrilateral (B) Special type of triangle (C) Parallelogram (D) Square

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6th Class Mathematics

116

III. Paragraph Type

  17. In the figure below, straight lines AS and HR   intersect at point O and ON  HR . L

H

N A O

S R

Using this information, find each of the following: (i) List all the possible acute angles. (ii) Write the number of right angles from the figure. (iii) Write two pairs of complementary angles.

IV. Integer Type 18. The number of sides of a regular polygon, if each of its interior angles is 1350 is given by ? 19. If each interior angle of a regular polygon is twice as large as the exterior angle, the number of sides are ? 20. The number of triangles with any three of the lengths 1, 4, 6 and 8 cms as sides is ? 21. State the number of right angles in a complete turn. 22. How many vertices of an hexagon 23. How many side of an octagon

V. Matrix Matching (Match the following) 24. Column – I (A) The point of intersection of attitude (B) The point of intersection of the medians of triangle (C) The point of intersection of Perpendicular bisectors of the Sides of the triangle (D) The point of intersection of Angular bisector of the angles of triangle

Column – II (p) Incenter (q) Circumcentre (r) Centroid (s) Orthocenter Of triangle.

25. In a Parallelogram

Column – I (A) Adjacent sides are equal (B) One angle is right angle (C) Diagonals perpendicular (D) Adjacent sides are equal and the Angle is 90°

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Column – II (p) Square (q) Quadrilateral (r) Rectangle (s) Rhombus

Mensuration

Learning Outcomes 

Perimeter



Area



Perimeter and Area of Plane Figures



Volume and surface Area of Cuboid and Cube

1. Perimeter Closed Figure: A figure with no open ends is a closed figure. Regular closed figures: A closed figure in which all the sides and angles equal.

Chapter – 6

By the end of this chapter, you will be understand

Equilateral Triangle: A triangle with all its sides and angles equal is called an equilateral triangle. The perimeter of an equilateral triangle with the side ‘a’ = a + a + a =3×a

Perimeter: Perimeter is the distance covered along the boundary forming a closed figure when we go round the figure once. The concept of perimeter is widely used in real life.

a

a

Eg: •

For fencing land. • For building a compound wall around a house. The perimeter of a regular closed figure is equal to the sum of its sides. Perimeter of a Rectangle: = Length (l)+Breadth (b)+Length (l)+Breadth (b) = 2(l + b)

l b

b l rectangle

Perimeter of a Square: =s+s+s+s =4×s s

s

s s square

a

2. Area The amount of surface enclosed by a closed figure is called its area. The following conventions are to be adopted while calculating the area of a closed figure using a squared or graph paper. 1. Count the fully-filled squares covered by the closed figure as one square unit or unit square each. 2. Count the half-filled squares as half a square unit. 3. Count the squares that are more than halffilled as one square unit. 4. Ignore the squares filled less than half. For example, the area of this shape can be calculated as shown:

118

6th Class Mathematics Number

Area estimate (sq. units)

Fully filled squares

6

6

Half-filled squares Squares filled more than half Squares filled less than half

6

7×½

7

0

0

0

Covered area

Area covered by full squares = 6 x 1 = 6 sq. units Area covered by half squares = 7 x ½ = 7/2= 3 ½ sq. units Total area of the given shape = 6 + 3 ½ sq. units Thus, the total area of the given shape = 9 ½ sq. Units Area of a rectangle can be obtained by multiplying length by breadth. Area of the square can be obtained by multiplying side by side.

3. Perimeter and Area of Plane Figures Triangle

Right Angle Triangle In a triangle ABC if one angle is 900, then it is called a right angled triangle A

d

h

B

1 1 base × height = bh = b d 2  b 2 2 2 Acute-Angle Triangle The triangle which is having all angles less than 900 is called an acute angle triangle. Area A =

A

The closed figure obtained by joining three non collinear points is called a triangle. Let a, b, c lengths of sides triangle then s = a + b + c is a perimeter of triangle, s 

C

b

Note: If ABC is a Right Angled Triangle then by Pythagoras Theorem (Hypotenuse)2 = (Side)2 + (Side)2 Perimeter of right angled triangle = b + h + d

abc is called 2

a

b

semi perimeters of the triangle h

Equilateral Triangle A triangle which is having equal sides is called an equilateral triangle.

B

D c

C

Obtuse Angled Triangle: In a triangle one angle is above 900 , then it is called an obtuse angle triangle.

A

A

b C

3a/2

B

a/2

a/2

c

a

C

B

Let the side of an equilateral triangle be ‘a’ then

Quadrilaterals

3 a 2 Perimeter of equilateral triangle = 3a units.

Rectangle The quadrilateral which is having pair of equal and parallel opposite sides and one angle is 900 .

height of equilateral triangle =

1 1 1 3 3 2 Area = base × height = a . a = a 2 2 2 2 4

d

A

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C

D

b

l

B

Mensuration

119

Let ABCD be a rectangle of length = l units, breadth = b units, diagonal = d units. • Perimeter of the rectangle = 2(l + b) units • Area = l × b units

Formative Worksheet 1.

• Area (A) = l d 2  l 2  b d 2  b 2 • Diagonal (d) =

2

l b

kitchen is of 18 ft length and 16 ft width and each square tile has an area of 4 ft2. What is the area of

2

Square The rectangle which is having all sides equal, all angles are right angles is called Square. A

a

B

a

d

a

C

a

D

Let the side length of a square be ‘a’ units, then • Perimeter of square = 4a units • •

Diagonal of the square = a   a 2  2a units Area of the square = a2 sq.units

•

Area of the square =

2.

3.

Side of square =

500 m What is the perimeter of the farm? (A) 250 m (B) 500 m (C) 1000 m (D) 2000 m A rectangular hockey field has length 118 yards and width 180 feet. What is the perimeter of the hockey field?

Area units

4.

The area of a rectangular playground is 300 m2. The length of the playground is 2,000 cm. The width of the playground is (A) 10 m (B) 15 m (C) 20 m (D) 25 m

5. b

118 Yards (B) 356 feet (D) 356 miles

(A) 356 m (C) 356 yards

Rectangular paths The path obtained between outer rectangular and inner rectangular fields is called Rectangular path.

w

the kitchen floor? (A) 268 ft2 (B) 278 ft2 2 (C) 288 ft (D) 298 ft2 Sherman has a triangular farm whose dimensions are shown in the figure.

1 × (diagonal ) 2 2

1 2 = d sq.units, where ‘d’ is length of diagonal 2 •

Sandra wants to get her kitchen floor tiled with square tiles such that each tile has a side of 2 ft. The

Zachary draws six different rectangles on a centimeter grid paper.

b + 2w

l l + 2w

If ‘l’, ‘b’ are length and breaths of inner rectangle and ‘w’ be width of path then • Length of outer rectangle = l + 2w • Breadth of outer rectangle = b + 2w • Area of inner rectangle = lb • Area of outer rectangle = (l + 2w) (b + 2w) • Area of path = (l + 2w) (b + 2w) – lb = 2(l + b)w + 4w2

A

B D C

F E The area of which two rectangles is the same? (A) C & D (B) B & D (C) B & E (D) C & E

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120

6th Class Mathematics

The area of a rectangular field is 225 m2. The width of the field is 400 cm. What is the length of the field in meters? (A) 46.25 (B) 56.25 (C) 66.25 (D) 76.25 7. The maximum area of a rectangle with integral lengths and having a perimeter 36 m is (A) 80 m2 (B) 81 m2 (C) 90 m2 (D) 91 m2 8. For a rectangle with integral lengths of sides and 196 m2 area, the minimum perimeter is obtained by taking the length of the rectangle as (A) 4m (B) 7m (C) 14m (D) 28m 9. In order to get the minimum perimeter of a rectangle of 225 m2 area and having sides of integral lengths, one of the sides of the rectangle should be of length (A) 3 m (B) 5 m (C) 11 m (D) 15 m 10. The minimum perimeter of a rectangle of area 324 6.

The perimeter of the figure PQRSTUP is (in cms) (A) 24 (B) 27 (C) 28 (D) 32 14. The area of a rectangle with length l and width w is 90 square units. Which of the following rectangles has the minimum perimeter and also satisfies the given condition? 6

(A)

5

(B)

15

18 9

(C)

3

(D)

10 30 15. If the length of the side of each small square in the given grids is 1 unit, then which of the following figures does not show a rectangle of perimeter 18 units?

m2 and having sides of integral lengths is (A) 36 m (B) 72 m

(C) 120 m (D) 170 m

11. The given figure shows a rectangle ABCD such that ÄEFG has been cut out of the rectangle. The remaining portion of the rectangle has been shaded. D

G

11 cm

E

A

The perimeter of the shaded region is (A) 52 cm (B) 59 cm (C) 72 cm (D) 79 cm 12. The given figure shows an equilateral triangle AED which shares a side AD with parallelogram ABCD. The lengths of adjacent sides AB and BC are 50 cm and 35 cm respectively. D

1.

C

S

P

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The length and width of a rectangle with integral sides and perimeter as 18 units, such that it has the maximum possible area, are respectively (A) 8 units and 1 unit (B) 7 units and 2 units (C) 5 units and 4 units (D) 6 units and 3 units The given figure shows four rectangles R1, R2, R3 , and R4, each of which has an area of 36 cm2 . 1 cm

B

The perimeter of the figure ABCE is (in cms) (A) 165 (B) 180 (C) 205 (D) 240 13. A rectangle of certain dimensions is chopped off from one corner of a larger rectangle as shown in the figure. The length of the side PQ = 10 cm and that of QR = 6 cm. U

(D)

Conceptive Worksheet

2. A

(C)

B

F

14 cm 20 cm

E

(B)

C

16 cm

10 cm

(A)

R T Q

3 cm

R1 36 cm R3 12 cm

2 cm R2 18 cm 4 cm

R4 9 cm

The perimeter of which of the given rectangles is minimum? (A) R 1 (B) R 2 (C) R 3 (D) R 4

Mensuration 3.

121

Veronica places her mathematics book on a two centimeter grid paper. The total area of the book is 200 cm2 .

8.

The given figure shows a window and its dimensions. 58 cm A

F

B

E

82 cm

4.

5.

How many squares does the book cover on the grid paper? (A) 25 (B) 50 (C) 100 (D) 200 The sides of a triangular pizza slice are equal to 20 cm. The area of the pizza slice is 173 cm2 . What is the perimeter of the pizza slice? (A) 30 cm (B) 60 cm (C) 80 cm (D) 120 cm Jane draws the alphabet L as shown in the figure.

82 cm C

D 121 cm

The estimated perimeter of the window is (A) 200 cm (B) 250 cm (C) 400 cm (D) 450 cm 9. A

29 m

39 m

B

39 m

2 cm

D 1 cm 3 cm

What is the perimeter of the alphabet L as drawn by Jane? 6.

(A) 10 cm (B) 12 cm (C) 14 cm (D) 16 cm The given figure shows a geometrical construction on a grid comprising squares of area 1 square unit each.

The area of the geometric figure is (A) 15square units (B) 16 square units (C) 17 square units

(D) 18 square units

48 m

C

The perimeter of the given figure is (A) 140 m (B) 150 m (C) 155 m (D) 165 m 10. Aubrey wants to buy tiles for the floor of her drawing room. The dimensions of the floor of her drawing room are shown in the figure.

5m

10 m The perimeter of the floor is (A) 10 m (B) 20 m (C) 30 m (D) 40 m 11. Aubrey wants to buy tiles for the floor of her drawing room. The dimensions of the floor of her drawing room are shown in the figure.

5m

7. A

51 cm

B 38 cm

38 cm D

51 cm

10 m What is the total area of the floor? (A) 40 m2 (B) 50 m2 (C) 60 m2 (D) 70 m2

C

The estimated perimeter of the given figure is (A) 120 cm (B) 140 cm (C) 160 cm (D) 180 cm www.betoppers.com

122

6th Class Mathematics

12. Achary draws six different rectangles on a centimeter grid paper.

A

B

The area of a parallelogram is the product of its base and perpendicular height or altitude. Any side of a parallelogram can be taken as the base. The perpendicular dropped on that side from the opposite vertex is known as the height (altitude). In the below figure, the area of parallelogram ABCD = AB × DE or AD × BF. D

C

D

C

D height

C E

Which rectangle has the maximum perimeter? (A) A (B) C (C) D (D) F 13. The given figure shows a polygon drawn on a grid paper. The length of the side of each small square in the grid is one unit. A B

F base

F

P O

N M

C D E

L K

A

B

E

B

A

base

height

A parallelogram in which the all sides are equal is called a rhombus. The perimeter and area of a rhombus can be calculated using the same formulae as that for a parallelogram. D

C

J

F G H

I

A

B

Circle What is the perimeter of the drawn polygon? (A) 16 units (B) 18 units (C) 24 units (D) 16 units 14. The given figure shows a polygon drawn on a grid paper. The length of the side of each small square in the grid is one unit. A B

F C

D

A circle is defined as a collection of points on a plane that are at an equal distance from a fixed point on the plane. The fixed point is called the centre of the circle. Circumference: The distance around a circular region is known as its circumference. Diameter: Any straight line segment that passes through the centre of a circle and whose end points are on the circle is called its diameter. Radius: Any line segment from the centre of the circle to its circumference.

E

What is the perimeter of the drawn polygon?

Parallelogram A quadrilateral in which both the pairs of opposite sides are parallel is called a parallelogram. The perimeter of a parallelogram is twice the sum of the lengths of the adjacent sides. In the below figure, the perimeter of parallelogram ABCD = 2(AB + BC) www.betoppers.com

Radius Circumference of a circle = 2r , Circumference of a circle =  d, where r is the radius of the circle or,, where d is the diameter of the circle.  is an irrational number, whose value is approximately equal to 3.14.

Mensuration

123

D D2 , if diameter D is given( r  ) 2 4

20.

Q

Area  d 4

2

Area  c 4 2

Formative Worksheet 16. The given figure shows a square piece of cardboard of side 10 inches. A rectangular piece of dimensions 3 inches × 2 inches is cut from the cardboard as shown in the figure.

R

17 cm

What is the area of the triangle PQR shown in the given figure? (A) 66 cm2 (B) 68 cm2 (C) 72 cm2 (D) 74 cm2 D

21.

C 5 cm

C2 , if circumference C is given ( c  D ) 4 Area = r2 r 0

P

8 cm

Circumference = Diameter x 3.14 Diameter(d) is equal to twice radius(r): d = 2r Circles with the same centre but different radii are called concentric circles. The area of a circle is the region enclosed in the circle. The area of a circle can be calculated by using the formula: r 2 , if radius r is given

A

B

E 12 cm

What is the area of the parallelogram ABCD as shown in the given figure? (A) 48 cm2 (B) 60 cm2 (C) 70 cm2 (D) 84 cm2 22. The height of a triangle is 2 cm less than twice its base. If the base of the triangle is 6 cm, then what is the area of the triangle? (A) 15 cm2 (B) 20 cm2 (C) 25 cm2 (D) 30 cm2 D 23. C 14 units A

B

E 25 units

What is the area of the parallelogram shown in the given triangle? (A) 200 unit2 (B) 250 unit2 2 (C) 300 unit (D) 350 unit2

10 in

3 in

2 in

Area of the remaining portion of the cardboard is (A) 90 inch2 (B) 92 inch2 (C) 94 inch2 (D) 96 inch2 17. If the perimeter of a 180 m wide rectangular field is 840 m, then the area of the field is (A) 42, 200 m2 (B) 43, 200 m2 (C) 44, 200 m2

A

24.

10 in

(D) 45, 200 m2

18. Sue measured the perimeter of the rectangular top of a coffee table which came out to be 12 ft. The length of the top of the coffee table is 4 ft. What is the area of the top of the coffee table? (A) 6 ft2 (B) 8 ft2 (C) 10 ft2 (D) 12 ft2 19. The base of a parallelogram is 3 cm more than twice the height of the parallelogram. If height of the parallelogram is 4 cm, then the area of the parallelogram is (A) 42 cm2 (B) 44 cm2 (C) 56 cm2 (D) 58 cm2

18 units

B

24 units

C

What is the area of the triangle shown in the given triangle? (A) 200 unit2 (B) 208 unit2 2 (C) 216 unit (D) 224 unit2 25. The given figure shows a parallelogram of 20 m height. The base of the parallelogram is 40 m long. 40 m

20 m

40 m

What is the area of the parallelogram? (A) 200 m2 (B) 400 m2 (C) 600 m2 (D) 800 m2 www.betoppers.com

124

6th Class Mathematics

26. The given figure shows a triangle of 18 m height. The base of the triangle is 20 m long.

Conceptive Worksheet 15. The given figure shows a quadrilateral ABCD. A 10 cm 7 cm

18 m

C B

20 m

What is the area of the triangle? (A) 180 m2 (B) 190 m2 (C) 200 m2 (D) 210 m2 27. The width of a rectangular field, whose length is 35 m and perimeter is 130 m, is (A) 25 m (B) 30 m (C) 35 m (D) 40 m

What is the area of quadrilateral ABCD? (A) 84 cm2 (B) 120 cm2 2 (C) 180 cm (D) 204 cm2 16. The given figure shows a rectangle ABCD. A

D

28. What is the area of a rectangular field, which has a perimeter of 240 ft and a length of 80 ft? (A) 3, 200 ft2

(B) 3, 400 ft2

(C) 3,600 ft2

(D) 3,800 ft2

A

29. 5 cm

90° B

C

10 cm

   

B

   

x  4 cm 2 

C

3 x 1 cm  4 

If perimeter of rectangle ABCD is 54 cm, then what is its area? (A) 140 sq cm (B) 152 sq cm (C) 162 sq cm (D) 170 sq cm 17. The given figure shows a quadrilateral ABCD such that AB = CD = 4 cm. The length of AC is 6 cm. A

What is the area of ABC , as shown in the given figure? (A) 15 cm2 (B) 25 cm2 (C) 40 cm2 (D) 50 cm2

B

A

30.

D

4 cm B

C

D 8 cm

What is the area of ABC , as shown in the given figure? (A) 12 cm2 (B) 16 cm2 (C) 28 cm2 (D) 32 cm2 31. The radius of a car wheel is 15 units

C

What is the area of the quadrilateral ABCD? (A) 10 cm2 (B) 12 cm2 (C) 20 cm2 (D) 24 cm2 18. The ratio of the circumference of two circles is 7:11. What is the ratio of the areas of the two circles? 19. A circular garden is surrounded by 3 m wide path. The circumference of the garden excluding the path is 88 m.

Garden

15

3m

What is the area of the wheel? (A) 125  (units)2 (B) 225  (units)2 2 (C) 325  (units) (D) 425  (units)2

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What is the area of the path surrounding the garden? (A) 143.28 m2

(B) 292.28 m2

(C) 706.28 m2

(D) 908.28 m2

Mensuration

125

20. In the given figure, BCDE is a square. ABC and EDF are right-angled triangles at its two edges. A

Cuboid A solid bounded by six rectangular plane faces is called cuboid

1.3 cm B

1.2 cm

E

4. Volume and Surface Area of Cuboid and Cube

C D

3.7 cm

Cuboid

The net belonging to cuboid is

F

What is the area of the given figure? (A) 2.20 m2

(B) 2.40 m2

(C) 3.14 m2

(D) 3.84 m2

21. How many squares of side 5 cm can be placed over a square of side 30 cm such that no two smaller squares overlap? (A) 24

(B) 30

(C) 36

(D) 40

22. The given figure shows a circle inscribed in a square of side 14 cm.

• Number of edges = 12 • Number of vertices = 8 • Number of faces = 6 flat rectangular faces Cube The cuboid whose length, breadth and height are equal is called a cube

h b l

What is the area of the shaded region? (A) 40 cm2 (B) 42 cm2 (C) 45 cm2 (D) 49 cm2 23. If the distance covered by a farmer around a rectangular park of length 180 m is 500 m, then what is the area of the rectangular park? (A) 11400 m2 (B) 12600 m2 (C) 13600 m2 (D) 14400 m2 24. The length and breadth of a rectangular cardboard are 65 cm and 50 cm respectively. If the length of the side of a square-shaped cardboard is 5 cm, then how many such square cardboards can be placed on the rectangular cardboard such that there is no overlapping of the square cardboards? (A) 108 (B) 112 (C) 120 (D) 130 25. How many squares of side 5 cm can be placed over a square of side 30 cm such that no two smaller squares overlap? (A) 24 (B) 30 (C) 36 (D) 40

cube The nets belonging to cube is

• Number of edges = 12 • Number of vertices = 8 • Number of faces = 6 flat square faces Volume The space occupied by a solid body is called its volume. The units of volume are cubic millimetres (mm3) or cubic centimetres (cm)3 or cubic metre (m)3 etc.....

Volume and Surface Area of Cuboid A solid bounded by six rectangular plane faces is called cuboid.

h b l

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126

6th Class Mathematics Let the Dimensions of cuboid be length = ‘l’ units, breath = ‘b’ units, height = ‘h’units, then • Diagonal of cuboid  l 2  b 2  h 2 units • Total surface Area of cuboid = 2( lb + bh + lh sq units • Lateral surface Area of cuboid = 2 (l × b)× h sq. units • Volume of cuboid = lbh cubic units • Area of 4 walls of room = 2 (l × b)× h sq.units

38. A rectangular cardboard sheet has four equal squares removed from its corners as shown in the figure. The obtained piece is then used to make a box. The base of the box has a length of 15 cm and a width of 12 cm. The volume of the box is 540 cm3 .

12 cm

Volume and Surface Area of Cube The cuboid whose length breath and height are equal is called a cube.

h

15 cm

What was the original area of the sheet of cardboard? (A) 251 cm2 (B) 270 cm2 2 (C) 315 cm (D) 378 cm2 39. The given figure shows a cube whose edges are equal. Its length, width and height are S units.

b l

s

cube Let the edge of a cube be ‘a’ units then • Diagonal of cube =

3 a units • Total surface Area of cube = 6a2 square units • Lateral surface Area of cube = 4a2 square units • Volume of cube = a3 cubic units

Formative Worksheet 32. What is the volume of a cuboid of dimensions 2.8 cm × 1.6 cm × 4 cm? (A) 17.92 cm3 (B) 22.08 cm3 (C) 35.84 cm3 (D) 44.16 cm3 33. A metallic cube is melted in the form of a cylinder in such a way that its radius is half the length of edge of the cube. What fraction of the length of the edge of the cube is the height of cylinder formed?

7 3 14 7 (A) (B) (C) (D) 44 22 11 11 34. In a cubical box of side 36 cm, exactly 216 small identical cubes can be placed. What would be the length of the side of each small cube? (A) 4 cm (B) 6 cm (C)12 cm (D)14 cm 35. If the volumes of two cubes are in the ratio 125: 343, then what is the ratio of the lengths of their edges? (A) 5:7 (B) 7:9 (C) 25:49 (D) 49:81 36. What is the length of the edge of the cube whose volume is 2744 cm3? (A) 14 cm (B) 12 cm (C) 7 cm (D) 6 cm 37. What is the total surface area of a cube whose volume is 1728 cm3? (A) 144 cm2 (B) 216 cm2 (C) 436 cm2 (D) 864 cm2 www.betoppers.com

s

The formula for volume of the cube is given by

Conceptive Worksheet 26. The adjacent figure shows a rectangular prism of length l units, width w units and height h units.

h

w l

The formula for the total surface area of the rectangular prism is (A) 2 lw + wh + hl (B) lw + 2(wh + hl) (C) 2 (lw + wh + hl) (D) lw + wh + hl 27. If the sides of a cube are doubled, then its volume becomes (A) One-fourth of the initial volume (B) One-half of the initial volume (C) Four times the initial volume (D) Eight times the initial volume 28. In a factory, metal sheets are molded into cubes. They are then kept in heating chambers. Due to expansion, the volume of cubes increases to 3 times the original volume.If ‘a’ is the edge of the original cube, then the edge of an expanded cube is (A)

1 a 3

1 (B) 3 a 3

(C) 3 3 a

(D) 3a

Mensuration 29. David’s room is 8 m long, 7 m wide and 9 m high. He wants to paint his room red. The cost of painting 1 m2 area is Rs.2.50. The cost of painting the room is (A) Rs.815 (B) Rs.825 (C) Rs.835 (D) Rs.845 30. Figure I shows a cube. Each side of the cube measures a units. It is then opened to obtain a twodimensional net, as shown in figure II.

a units a units

a units

Figure I a

a

1

a a

2

3

a a

a 4

a

a

5

6

a

a

127 What is the total surface area of the rectangular prism? (A) (lw + wh) unit2 (B) (lw + lh) unit2 (C) (lw + wh + hl) unit2 (D) 2(lw+wh+hl) unit2 32. If the length of the edge of a cube is 3.2 cm, then the total surface area of the cube is (A) 61.44 cm (B) 61.44 cm2 (C) 61.44 cm3 (D) 61.44 cm4 33. The volume of a rectangular prism 10 dm × 4 dm × 2 m is (A) 800dm (B) 800dm2 (C) 800dm3 (D) 800 dm4 34. A rectangular block of silver measures 13 cm × 10 cm × 8 cm. It is given that 1 cm3 of silver weighs 10.5 g. The weight of the silver block, in kg, is (A) 5.46 (B) 10.92 (C) 54.64 (D) 86.42 35. The given figure shows a small rectangular prism. The dimensions of the rectangular prism are also shown in the given figure.

a 2.7 cm

a

1.2 cm

Figure II What is the total surface area of the given cube? (A) a2 unit2 (B) 2a2 unit2 (C) 4a2 unit2 (D) 6a2 unit2 31. Figure I shows a rectangular prism of length l units, width w units, and height h units. It is then opened to obtain a two-dimensional net, as shown in figure II.

6.9 cm

The difference in the estimated surface area and the actual surface area of the rectangular prism is (A) 4.7 cm2 (B) 3.7 cm2 (C) 2.7 cm2 (D) 1.7 cm2 36. A cuboidal tank is 7 m long, 3 m wide and 5 m high. Water is stored in the tank. The maximum quantity of water that can be stored in the tank is (A) 63 m3 (B) 75 m3 (C) 105 m3 (D) 147 m3

Summative Worksheet 1.

h

The perimeter of a shaded figure is A

l

L 2m K

VI l

I

3m

J

E

F ?

1m

h

w IV h

III l V

D 2m C

3m

Figure I

w I h

B

1m

w

II l

5m

H

2.

h w

3.

G

The perimeter of a rectangular plot is 100m and its three sides are 35.3m, 25.2m,18.5m, then the length of a fourth side of a rectangular plot ? The side length of a regular decagon is 13m, then the perimeter of a regular decagon ?

Figure II www.betoppers.com

128 4.

6th Class Mathematics Area of the shaded portion of the following figure A

13. Little Johnny got a cube as a birthday gift. All the six faces of the cube are of different colours and each side of the cube is of 5 cm length.

10 m 5m

F

5m

G D B

5.

6.

7. 8.

?

E 15 m

C

What is the capacity of the cube?

The area of a rectangular plot is 50625 Sq.m and its ratio of the length and breadth of a plot is 5.3, find the perimeter of a rectangular plot ? There are two fields one is a rectangle whose length and breadth is 35m and 20m and second field i s square of whose side is 55m, find the ratio of their areas, perimeters ? Find the area of a Rhombus one side of which measure 15m and one diagonal 17m ? A path of 2 meters wide runs around and inside a rectangular plot of measurements 25m and 20m, the cost of levelling the path at the rate of Rs.1.25 per square meter ?

(B) 0.001 25 m3

(C) 0.012 5 m3

(D) 0.125 m3

14. A rectangular prism is made by joining some cubes as shown in the given figure. The volume of each small cube is 1 cm3 .

1 cm

The volume of a rectangular prism is

25 m

(A) 32 cm3 (B) 36 cm3 (C) 42 cm3 (D) 46 cm3

2m 21 m

20 m

HOTS Worksheet 2m

2m

(A) 0.000 125 m3

1.

The figure shows a rectangle 40 cm wide and 0.6 m long.

2m

0.6 m

1 How many marble stones of length 5 m and 2 breadth 1m can be spread in a room of length 13m and breadth 11m ? 10. If the perimeter of a rectangle and a square, each is equal to 100m and the difference of their areas is 225, find the sides of the rectangle ? 11. The area of a rhombus is 132 cm2. If length of one of its diagonals is 11 cm, then what is the length of the other diagonal? (A) 12 cm (B) 14 cm (C) 22 cm (D) 24 cm 12. Theodore uses a non-stretchable, closed string of length 10 cm to construct a rectangle on a geoboard as shown in the figure. 9.

A

40 cm

What will be the area of the rectangle in sq. cm? (A) 24 2.

(B) 240

(D) 2800

The figure shows rectangle PQRS of 10 cm length and 8 cm width. P

10 cm

S

8 cm

B Q

2 cm

2 cm D

(C) 2400

C

What is the length of the sides AB and DC? (A) 1 cm (B) 2 cm (C) 3 cm (D) 4 cm www.betoppers.com

R

What is the perimeter of the given rectangle? (A) 36 cm (B) 38 cm

(C) 40 cm

(D)42 cm

Mensuration 3.

129

The given figure shows a rectangle ABCD of 12 cm length and 8 cm width, drawn on grid paper. Each square grid on the gird paper has a side of 1 cm and an area of 1 cm2 . A

9.

Smita has a garden which is in the shape of the given figure. What is the perimeter of the garden? 10.5 m

D

7m

10.

B

C

The area of rectangle ABCD is (A) 92 cm2 (B) 94 cm2 (C) 96 cm2 (D) 98 cm2 4.

11.

A rectangular sheet of paper is 10 cm long and 14 cm wide. It is folded once as shown in the figure (i) and then folded again as shown in the figure (ii). 12.

7cm 10cm

10cm

14cm (i)

5.

6.

7.

(ii)

What will be the area of the rectangle formed after the second fold? (A) 55 cm2 (B) 45 cm2 (C) 35 cm2 (D) 25 cm2 The perimeter of a 100 cm long and 30 cm wide towel will be (A) 270 cm (B) 260 cm (C) 250 cm (D) 240 cm What is the area of a rectangle, which is 5 cm long and 10 cm wide? (A) 20 cm2 (B) 30 cm2 (C) 40 cm2 (D) 50 cm2 What is the area of the shaded region in the given figure? 2cm 14cm

13.

18 25

The volume of the figure is (A) 2,050 ft3 (B) 3,625 ft3 3 (C) 4,050 ft (D) 5,625 ft3

IIT JEE Worksheet I.

Single Correct Answer Type

1.

If the perimeter of a rectangle is 1024m and its length is 250m then its breadth is ? (A) 40.96m (B) 262m (C) 774m (D) 256000m If the side of a square is 22.5cm then its perimeter is ? (A) 0.90m (B) 5.625m (C) 90m (D) 506.25m If the regular octagon having side length is 14.5m, then its perimeter is ? (A) 1.8125m (B) 22.5m (C) 116m (D) 130.5

2. 10cm

8.

9

9

2cm

(A) 240 cm2 (B) 250 cm2 2 (C) 268 cm (D) 274 cm2 What is the area of a circle whose circumference is 110 cm? (A) 468 cm2 (B) 625 cm2 2 (C) 861.4 cm (D) 962.5 cm2

(A) 27 m (B) 39 m (C) 50 m (D) 64 m A circular garden of diameter 21 m is surrounded by a footpath of width 1.4 m. What is the area of the path? (A) 74.12 m2 (B) 98.56 m2 2 (C) 105 m (D) 121 m2 A concrete truck, holding 7.8 cubic yards of concrete, arrives at a work site. A patio 18 feet wide and 4 inches thick has to be constructed. The length of the patio, if the entire concrete is used, will be (A) 15 feet (B) 20 feet (C) 35 feet (D) 40 feet A die is a cube, molded from hard plastic. The edge of a die measures 0.62 inches. A mould can make 100 dice at once. The volume of plastic needed to fill the large mould is (A) 62 cubic inches (B) 30 cubic inches (C) 24 cubic inches (D) 12 cubic inches Use the following information to answer the next question.

3.

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130 4.

5.

6th Class Mathematics The perimeter of the parallelogram, whose adjacent sides 8.7m and 5.3m is ? (A) 14m (B) 44.52m (C) 1.64m (D) 28m Perimeter of the following closed figure is ?

8 cm

4 cm

4 cm

1 cm

(A) Area of circle =

2 cm 2 cm

6.

7.

8.

(A) 24cm (B) 21cm (C) 23 cm (D) 25cm If the area of a rectangular field is 562.5 sq.m and it’s length is 22.5m, then the breadth is ? (A) 25m (B) 12656.25m (C) 258.75m (D) 540m The ratio of the length and breadth of a rectangular field is 5:6, if the length is 80m then the area is ? (A) 90 Sq.m (B) 340 Sq.m (C) 7680 Sq.m (D) 192000 Sq.m If the area of a square is 1024 Sq.m then length of the square is ? (A) 1048576 m (B) 32 m (C) 256 m (D) 4096 m

II. Multi Correct Answer Type 9.

12. Each side of a square park measures 95 m. The distance covered by a person going round the park 6 times is: (A) 570 m (B) 2,280 m (C) 6  4  95 m (D) 6  95 m 13. Which of the following statements is correct?

What is the area of the triangle shown in the figure?

circumference  Radius 2

1 (B) Area of triangle =  Base  Perimeter 2 (C)

 perimeter of square  Area of square=

2

16 (D) Area of rectangle = 2 x Area of Square

III. Paragraph Type 1 × base × height 2 (ii) Area of rectangle = Length × breadth (iii) Area of square = side × side (iv) circumference of a circle = 2 ×  × Radius Use the above information to solve the following questions 14. Find the area of triangle ABC. (i)

Area of triangle =

20 cm C

A 3 cm 14 cm

20 cm

B

12 cm

21 cm 2

(A) 126 cm

(B) 0.0126 m2

1  21  12 cm2 2 10. Which measurement of a circle will give the length of its diameter? (C) 12600 mm2

(A) 2  Radius

(D)

(B)

2  Area Radius

15. Find the area of the shaded part. 5 cm A 4 cm B 1 cm X 2 cm 2 cm

Y

3 cm

D

3 cm 6 cm 16. Calculate the circumference of a circle with a radius of 3.2 cm. (  = 3.142)

IV. Integer Type

17. Find the side of the square whose perimeter is 2  Circumference 2  Area 20 m. (C) (D) 2π circumference 18. A piece of string is 30 cm long. What will be the 11. Krishna made a drawing of his rectangular kitchen length of each side if the string is used to form a for art class. The length of the drawing was 8 cm, regular hexagon? and the width of the drawing was 2 cm less than the 19. The area of a rectangular garden 50 m long is length. Find the perimeter of the drawing. 300 sq m. Find the width of the garden. (A) 28 m (B) 0.28 m 20. A table-top measures 2 m by 1 m 50 cm. What is its (C) 28 cm (D) 2  (8 + 6) cm area in square metres?

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Mensuration

131

21. Find the area of each of the following triangles:

4 cm

3 cm

2 cm 4 cm (a)

3 cm (c)

(b)

3 cm (d)

V. Matrix Matching (Match the following) 22.

Column – I (A) The perimeter of a rectangle is

Column – II (

)

(p) 90

(

)

(q) 40

(

)

(r) 35

(

)

(s) 25

(

)

(t) 36

(

)

(u) 45

150m, if the length is 40m, then breadth is (B) The perimeter of a square is 180m, then the length of the side is (C) The perimeter of a regular nanagon is 360m, then the length of the side is (D) The area of a square is 625 Sq.m, then length of the side is

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6th Class Mathematics

By the end of this chap ter, you will unde rstand  Fundamental Concepts

Chapter - 7

Data Handling

Learning Outcomes

 Pictorial Representation of Data

1. Introduction Sir Ronald Fisher introduced the concept of data handling or statistics. Indian mathematicians P.C. Mahalanobis and C.R. Rao have also played major role in the field of statistics. It is being used in all the fields like planning and projects, government budgets, population analysis, share market, student data analysis etc. Today it has developed as special branch of mathematics. Statistics can also be termed as data handling. It is a process of drawing facts from numerical data. It includes collection, presentation and interpretation of the data.

2. Fundamental Concepts Data Gathering information in the form of number or numerical figures is called data. Eg: • Number of boys and girls in a school • Details of rain fall in various towns • Information about population • Students information in various categories • No. of vehicles produced in different years etc.... Advantage of Collecting Data in Numerical Form • It is easy to separate data into particular groups or categories • It is easy to analyse and interpret • It is easy to identify and find the values of required information.

Tally marks Tally marks are used to organise the observations. Record every observation by a vertical mark, but every fifth observation should be recorded by a mark across the four earlier marks, like this  We depict each observation with the help of tally marks.

For Example, we have a group of persons and their sizes of shoes. The tabular form representing the tally marks is as shown here. Size of Shoes 5 6 7 8 9

Tally Marks        

Number of persons 5 8 10 7 2

Range of Data The difference between the maximum and minimum values of given data is range of data. Eg: The marks obtained by 10 students of a class in mathematics in a unit test are as follows: 25,25,24,20,18,15,10,5,9,22 The highest mark = 25 and the lowest mark = 5 Therefore the range of marks = 25 - 5 - 20

Raw Data or Ungrouped Data If the collected information is presented randomly then it is called raw data. i.e, collection of observation gathered initially is called a composite raw data. Eg: The marks obtained by 30 students of a class in maths are as follows: 29,24,50,10,15,2,59,36,74,37,93,45,63,52,36,41,54, 37,83,51,29,36,51,47,83,78,88,47,41,91. The above given initial or original form of data is called raw data. In above example queries like: • How many have failed in exam • How many got above 70 marks • No. of students with average or good performance in the exam are difficult to analyse Disadvantages of Raw or Ungrouped Data • If no. of values are more, it is very difficult to analyse raw data. • Identifying different categories of data in raw data is time consuming.

6th Class Mathematics

134

Grouped Data If the raw data is classified or divided into groups or classes according to the requirement then it is called grouped data. Grouped data includes the terms; class interval, length of the class (or size of class interval) and frequency. Class Interval Dividing data into groups or intervals called class interval contains minimum and maximum value. These values are called as limits. Each class has lower limit and upper limit. Lower and Upper Limit of a Class The starting and end values of each class are called lower limit and upper limit respectively of that class. Eg: If the class interval is 1-20, then the lower limit of class is 1 and upper limit of class is 20. Note: The mid value of a class interval is called its class mark. Class Boundaries The average of upper limit of a class and the lower limit of the succeeding class is called upper boundary of that class. The upper boundary of a class becomes the lower boundary of that next class. Length or Size of the Class The difference between the upper and the lower boundary of a class is called length of the class or size of the class. Eg: Class intervals of marks and no. of students in each of the categories are as follows. Marks

Frequency

1 – 20

4

21 – 40

8

41 – 60

12

61 – 80

40

81 – 100

25

In above table: • 1 – 20, 21 – 40, 41 – 60, 61– 80, 81 – 100 are called class intervals. • 1, 21, 41, 61, 81 are lower limits and • 20, 40, 60, 80, 100 are called upper limits. Procedure to find Size of Class Upperbound of class (1 – 20) = Avg (upper limit of class 1 – 20 and lower limit of class 21 – 40)

20  21  20.5 2 www.betoppers.com 

Upper bound of class 1 – 20 = 20.5 Hence lower bound of class = (21 – 40) = upperbound of class 1 – 20 Now upperbound of class(21 – 40) = Avg(upper limit of class 21 – 40 and lower limit of class 41 – 60)

40  41  40.5 2 Hence upper bound of class(21 – 40) = 40.5 Now the length of the class(21 – 40) = Upperboundary of (21 – 40) – lower boundary of (21 – 40) = 40.5 – 20.5 = 20 Therefore length of the class 21 – 40 = 20 Since the length of each class interval should be equal. Therefore lengths of class or size of class of given table. Frequency of a Class 

The number of times a particular observation (or value) occurs in each class interval is called frequency of a class. When dividing data into grouped data, a line drawn for each observation in a class interval. These lines are called as “tally marks”. It is denoted by ‘1’. The number of tally marks in each group gives the number of observations in that class interval. This is called frequency of marks in each class will be the frequency of that class. Steps followed when preparing the Grouped Data Step-1: Find the range Step-2: The lowest and highest values of data should be covered in the classes Step-3: The length of class should be the same for all classes Eg: • 1-10, 11-20, 21-30,...... • 10-14, 15-19, 20-24, .....etc Taking classes like 1-9, 9-15 is wrong Step-4: Tally marks should be marked for each class. After every 4 tally marks 5th tally mark should be crossed for convenient of counting Advantages of Grouped Data • Collection of data in the form of numerical figures • The data can be grouped and presented with clarity • It is easy to analyse and interpret the data • New discoveries can be made and estimation can be done

Data handling

135

Arithmetic Mean Arithmetic mean is a number that lies between the highest and the lowest value of data. 1. Note: that we need not arrange the data in ascending or descending order to calculate arithmetic mean.

Formative Worksheet

Mode Mode refers to the observation that occurs most often in a given data. The following are the steps to calculate mode: Step-1: Arrange the data in ascending order. Step-2: Tabulate the data in a frequency distribution table. 2. Step-3: The most frequently occurring observation will be the mode.

Median Median refers to the value that lies in the middle of the data with half of the observations above it and the other half of the observations below it. The following are the steps to calculate median. Step-1: Arrange the data in ascending order. Step-2: The value that lies in the middle such that half of the observations lie above it and the other half below it will be the median. The mean, mode and median are representative values of a group of observations or data, and lie between the minimum and maximum values of the data. They are also called measures of the central tendency.

The given tally chart pertains to the favorite seasons of some people. Season Tally Marks Winter  Spring  Summer   Fall  The number of people who choose either fall or summer is (A) 6 (B) 8 (C) 10 (D) 12 The given table shows the numbers of students in a class, who play different musical instruments: Piano

5

S axophone

6

Trumpet

4

Guitar

10

Violin

7

Which frequency table correctly displays the given information? (A) Piano

//////

(B) Piano

Saxophone

////

Saxophone

//// /

Trumpet

//////

Trumpet

//// /

Guitar

///

Guitar

//// //

Violin

//////////

Violin

//// /

(C) Piano

(D) Piano

////

Saxophone //// /

3.

////

Trumpet

////

Guitar

//// ////

Violin

//// //

///

Saxophone

////

Trump et

////

Guitar

//// ////

Violin

//// /

The marks of 10 students studying at Oxford University, in Mathematics and Statistics, are given below. Marks in Mathematics :

25

40

30

35

21

45

23

33

10

29

Marks in Statistics:

30

39

23

42

20

40

25

30

18

19

What are the respective ranges of Mathematics marks and Statistics marks? (A) 35 and 20 (B) 30 and 24 (C) 35 and 24

(D) 30 and 20

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6th Class Mathematics

136 4.

5.

6.

7.

The number of candidates who appeared for a certain competitive exam in consecutive eight years is 12464, 14731, 11024, 13213, 14219, 15613, 11287, and 14162. What is the range of the number of candidates who appeared? (A) 5143 (B) 4589 (C) 3142 (D) 2764 What is the mean of the first six prime numbers greater than 34? (A) 48.8 (B) 45.3 (C) 44.6 (D) 43.5 The median of a set of numbers arranged in ascending order is 41, 60, 65, m – 2, 82, 85, 96 is 68. What is the value of m? (A) 64 (B) 68 (C) 70 (D) 72 The marks (out of 25) obtained by students of a certain class are 11, 12, 14, 13, 17, 11, 12, 12, 17, 21, 24, 14, 11, 13, 12, 17, 21, 19, 19, 24, 14, 15, 12, 17, 15, 19, 21, 13, 16, and 15. What is the mode of marks obtained by students of the class? (A) 12 (B) 14 (C) 15 (D) 17

2.

Age-group Number of people Number of who go to work people who play 10-13 / //// //// //// // 13-16 // //// //// //// //// /// 16-19 //// / //// //// / 19-21 //// //// / //// 22-24 //// //// //// /// 25-28 //// //// //// //// //// //

The ratio of the number of people who play between the age-group of 10 –13 to the number of people who go to work between the age-group of 25 – 28 is: (A) 19:23 (B) 7:10 (C) 17:25 (D) 18:29 A fair die lands on ‘1’ eight times, on ‘3’ five times and on ‘5’ seven times. Which of the following telly charts shows these results accurately? (A) Die res ults (B) Die res ults 1

//// ///

1

//// //

3

////

3

//// /

5

//// //

5

//// /

(C) Die res ults

4.

5.

6.

7.

Conceptive Worksheet

1.

3.

(D) Die res ults

Size of the bat

10

15

20

30

40

50

Cost

80

120

150

250

400

600

What is range of bat size? (A) 40 (B) 50 (C) 60 (D) 70 The distances (in m) covered by 12 participants in an interschool long jump competition are 3.4, 2.7, 3.2, 2.9, 4.1, 4.3, 3.8, 3.1, 4, 3.5, 3.9, and 4.4. What is the range of distances covered by the participants? (A) 2.6 m (B) 2.2 m (C) 1.9 m (D) 1.7 m The mean of seven numbers is 26. The sum of first four and last four numbers is 96 and 114 respectively. What is the fourth number? (A) 26 (B) 28 (C) 30 (D) 32 Consider the data:; 30, 16, 27, 16, 27, 30, 10, 30, 16, 27, 30, 10, 16, 27, x, 10 If the mode of the given data is 30, then what is the value of x? (A) 10 (B) 16 (C) 27 (D) 30 What is the median of the data 32, 37, 25, 42, 35, 136, 90, 45, 100? (A) 35 (B) 38 (C) 41 (D) 42

3. Pictorial Representation of Data The numerical data is represented through pictures or diagrams then it is called pictorial representation of data. The pictorial or visual representation for easy understanding a given data is called graph. We have different types of graphs. • Picture graph or pictographs • Bargraph or bar diagrams or column graph • Pie graph or pie diagrams • Line graphs

Pictographs Graphs which use pictures of objects or parts of objects is called pictograph or pictogram. In representation of data each picture represents only one object. • In a pictograph some times a symbol of picture or object may represents multiple units. • A rule that one picture of a object represents more objects is called scale of pictograph. Eg: 1

= 100 books

1

//// /

1

/

If it is half book like

3

//// //

3

///

= 50 books.

5

//// ///

5

////

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whose value is

Data handling

137

Steps to Follow while drawing Pictographs: • All the pictures should be of same size • The scale should be selected carefully to suit our 8. requirement • If half the picture is being used, the details should be mentioned clearly • The picture should be neat and attractive • The diagram should have a suitable and short heading Eg: The following pictograph shows the number of absentees in a class of 30 students during previous week. Days Monday Tuesday Wednesday

Number of absentees

  

Saturday

Scale

Type of shoes

Number of children wearing them

Lace-up shoes

Sneakers

Keds

 

= 1 – Absentees

On which day were the maximum number of students absent. (ii) Which day had full attendance. (iii) What was the total no. of absentees in that well Sol: (i) Maximum absentees were in saturdays since there are 7 pictures in the row for saturday. (ii) No one is absent on thursday, since there is no picture against thursday (iii) The total number of absentees in that week was 20. Since there are total 20 pictures. Disadvantages of Pictographs • Drawing pictographs is difficult and time consuming • Showing the whole information through pictograph is not possible always. = 100 books. Then we can

easily represents 50 books =

The fraction of children wearing lace-up shoes to those who wear the rest is 2 1 3 4 (A) (B) (C) (D) 9 5 10 15 9.

(i)

Eg: If 1

A group of children carried out a survey about different types and brands of shoes. The data was collected and written in the form of a table. Each student in the table represents ten students.

Ballerina shoes

Thursday Friday

Formative Worksheet

A survey was conducted across 66 sixthgrade students in order to know their favorite flavors of ice-cream. The information that was captured in the survey was then displayed in the form of a pictograph as: Chocolate Strawberry Fruits and nuts Vanilla

Key: Each

=?

In the key, how many students does each figure of the ice-cream cone represent? (A) 3 (B) 7 (C) 11 (D) 15

but it is difficult

to represent 46 books. Since there is no picture to represent 46 books. www.betoppers.com

6th Class Mathematics

138 10.

The given pictograph shows the number of oranges purchased by a group of four students. Student

12.

Number of oranges

Ben

Nina, George, Lucky, and Lara went out to pick seashells. They collected them in small bags that could hold 25 seashells each. The following pictograph shows the number of bags of seashells collected by each of them. Each bag has 25 seashells. Name Number of Marbles

Tim

Nina

Pam

George

John

Lucky

Key: Each

represents 3 oranges Lara

What is the difference between the number of oranges purchased by Pam and that purchased by Ben? (A) 7 (B) 9 (C) 11 (D) 13 11.

The following pictograph shows the number of motorcycles parked at a parking place on 7 days of a week. Day

Number of motorcycles parked

Sun Mon Tue Wed Thurs

What is the difference between the numbers of seashells collected by Nina and George?

Conceptive Worksheet 8.

A survey is conducted on a few people to know the kinds of books and magazines they read. The data is organized in the given pictograph. Book/ Magazine

Number of votes

Fashion magazines Drama

Fantasy

Fri Sat

What was the total number of motorcycles parked on all the seven days combined? (A) 100 (B) 103 (C) 108 (D) 112

Sports fiction Biographies Historical fiction

Which kinds of magazines/books are preferred by the maximum number of people? (A) Drama (B) Fantasy (C) Biography (D) Historical fiction

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Data handling 9.

139

The number of bananas eaten by a monkey on four days is given in the form of a pictograph:

10.

The given pictograph shows the number of flowers sold by a florist on a particular day. Flower

Number of flowers

Mon

Rose

Tue Wed

Tulip Thurs The monkey ate a total of 12 bananas over the given period.

Orchid

The total number of bananas eaten by the monkey is multiplied by four. The resulting number is (A) 24 B) 48 (C) 76 (D) 92

Lily

Key : Each

represents 9 flowers

The total number of flowers sold by the flower seller that day is (A) 192 (B) 194 (C) 196 (D) 198 11.

Types of trees in an orchard

Number of trees

1

Oak

30

Oak

2

Birch

50

Birch

3

Maple

30

Maple

4

Red bud

80

Red Bud

The following pictograph represents the given data:

???

The number of leaves corresponding to the Birch tree is

12.

(A)

(B)

(C)

(D)

The given pictograph shows the number of sunny days in the months of June, July and August of a particular year. Month

Sunny Days

June July August Key : Each

= 4 sunny days

The total number of sunny days in 3 months was (A) 12 (B) 24 (C) 36 (D) 48

Bargraph or Column Graph Representing data using pictographs is not possible in all cases, some otherway of representing data is bar diagrams or column diagrams. Representing the data with the help of bars or representing in a diagram is called a bar graph or bar diagram. • Each bar represents only one value of the data, hence there are as many bars as there are values in the data. Therefore no. of bars = no. of items • While drawing bar graph, the line drawn vertically is called y-axis and the line drawn horizontally is called x-axis • All bars should rest as same line called the base either on x-axis or y-axis www.betoppers.com

6th Class Mathematics

140 • The bars can be drawn either horizontally or vertically. The bars which are having base as xaxis are vertical bars and the bars which are having base as y-axis are horizontal bars • The length of the bar represents the values of the item • The (breadth) width of the bar does not represent any item. So that the width of all the rectangles (bars) is to be same for attractive graph • The distance between any two bars should be the same • The bars can be shaded with dots, lines or colours to make them attractive • While drawing bargraph the original values of the data cannot be shown in the graph. So that 1cm will be taken as a few units which is called scale of bargraph Eg: The marks obtained by bhargav in his annual examination are given as Subject

Marks obtained

Hindi

65

English

70

Maths

85

Science

55

Social st udies

60

60  6cm 10 Now we can draw bargraph for above data using step1 to step 4 No. of marks in Social  60 

100  90 

Marks obtained

80  70  60  50  40  30  20  10  Hindi English Maths Science Social Subjects

Formative Worksheet 13.

65  6.5cm 10

No. of marks in Mathematics = 85 

A fisherman recorded the number of different kinds of fish near the surface of the water. Tropical

7

Seahorses

4

Shark

1

Dolphins

3

Salmon

5

Which of the following bar graphs shows the given information correctly? (A)

7 6 5 4 3 2 1 Tropical Seahorses Shark Dolphins Salmon

Similarly no. of marks in English = 70 =

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55  5.5cm 10

Advantages of Bargraph • Bar diagrams are simple to draw • Bar diagrams provide easy comparison of the given data.

then corresponding bar diagram can be drawn as follows. Steps in drawing BarGraph • Draw a horizontal and vertical lines which are named as x-axis and y-axis on a graph paper. • Take the scale on y-axis as 1cm = 10 marks • Take marks obtained along y-axis and subject names on x-axis • Draw rectangles corresponding to given data. While drawing rectangles each rectangle should have same width and distance between any two rectangles should be equal. • Now we can find length of each bar No. of marks in hindi = 65 As per scale 10 marks = 1cm, Therefore 65 marks 

No. of marks in Science 

70  7cm 10

85  8.5cm 10

Data handling (B)

141 15. The given graph shows the number of designer clothes sold at a shop during a particular week.

7 6 5

y

4 3

225

2

200

1

175

Tropical Seahorses Shark Dolphins Salmon

(C)

150 125

7

100

6

75

5

50

4

25

3

0

2 1 Tropical Seahorses Shark Dolphins Salmon

7 6 5 4

50 

3

40 

2 1 Tropical Seahorses Shark Dolphins Salmon

14. The given graph shows the number of visitors to an amusement park for four months.

20  10 

(4)

(5)

(6)

(7)

Grade

Which grade has the least number of students? (A) 4 (B) 5 (C) 6 (D) 7 17. Read the following bar graph and answer the next question.

2400 2300 2200 2100 2000

Number of Shoes 

Number of visitors

30 

0

y

1900 1800 1700 1600 1500 0

Sat

The ratio of the minimum value of sale and the maximum value of sale is (A) 2 : 3 (B) 1 : 3 (C) 3 : 4 (D) 4 : 5 16. The given bar graph shows the number of students in grades 4, 5, 6, and 7 of a particular school. No. of students

(D)

x Mon Tues Wed Thurs Fri

x Jan

Feb

Mar Month

Apr

What can be predicted about the number of visitors in May? (A) They will increase. (B) They will decrease. (C) They will be the same as in April. (D) They will be the same as in February.

1000  900  800  700  600  500  400  300  200  100  3

4

5 6 8 7 Shoe Number 

9

10

What is the total number of all the shoes put together? (A) 2 900 (B) 3 000 (C) 3 200 (D) 3 400

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6th Class Mathematics

Conceptive Worksheet The given table shows the number of ice-cream cones sold in an ice-cream parlor during six days of a particular week: Day

Number of ice-cream cones

Monday

350

Tuesday

325

Wednesday

375

Thursday

400

F riday

425

Saturday

450

375 350

Mon Tue

425

Sat

200 150 100 50 0

375 350

Wed Thu Days

Fri

250

400

Mon Tue

Wed Thu Days

y

Fri

Sat

450

Oil

Steel Coal Gold Commodity

x

As seen in the given graph, the combined revenues of which two commodities amount to $350,000 in the particular month? (A) Oil and gold (B) Coal and gold (C) Oil and coal (D) Steel and coal 15. The given bar graph shows the number of vehicles passing through a road crossing in Texas at different time intervals on a particular day.

425 y

400 375

450

350

400

325 0

350 300

Mon Tue

Wed Thu Days

Fri

Sat

250 200 150 100

450

50 x

13-14h

12-13h

11-12h

400

10-11h

0

9-10h

425 8-9h

Number of ice-cream cones Number of ice-cream cones Number of ice-cream cones

(C)

400

14. The given graph shows the revenues earned by a country in a particular month. These revenues are related to the exports of oil, steel, coal, and gold by the country.

450

325 0

(B)

425

325 0

Which of the following graphs correctly represents the given set of data? (A)

450

Revenue ($ thousands)

13.

(D)

Number of ice-cream cones

142

375

The total number of vehicles passing through the crossing during the day is (A) 1,000 (B) 1,250 (C) 1,450 (D) 1,500

350 325 0

Mon Tue

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Wed Thu Days

Fri

Sat

Data handling

143

16. The graph in the given figure shows the number of trees planted during four consecutive years in a particular city.

=

Value of one components  observation  Aggregate of all observations

600 500 400 300 200

Item

100 0

 360

Step-2: Draw a circle of suitable radius to get the angles corresponding to different components Step-3: Write the title either on top or at the bottom Eg: Santosh earns Rs. 12000 per month his expenditure on various items during a month is as follows:

700

Number of trees planted

Sectorial angle for a given observations

2003

2004

2005 Years

2006

As seen in the given graph, the maximum number of trees was planted in the year (A) 2003 (B) 2004 (C) 2005 (D) 2006 17. The bar graph below shows the sales of almonds at a seven-eleven store in the U.S. for a particular year.

Amount spent

Found

Rs 2500

House rent

Rs 1800

Bike M aintenance

Rs 2400

Savings

Rs 3000

Misc

Rs 2300

Pie-chart to represent the above data as follows Step-1: Aggregate of all components = 12000 Sectorial angles corresponding to given data items are  2500   360   75  12000 

Sectorial angles for food  

Jan Feb Mar Apr May Jun July Aug Sep Oct Nov Dec

In which of the following months is the sale of almonds continuously decreasing? (A) January to May (B) March to August (C) May to September (D) June to October

PieGraph or Pie-Diagram In a pie-diagram, each observation is represented by the sector of a circle. The circle as a whole represents the total of the components. The pie diagram is drawn by first drawing a circle of suitable radius and then dividing the angle of 3600 at its center in proportion to the values of the various components. The areas of various sectors are in proportion to the angles which they make at the centre of the circle. Thus the areas of the sectors made from the circles are in proportion to the values of the components. The pie-diagram is advantageous to draw when we wish to compare items of the same form or when the absolute values in the data are not given but only proportional or percentage values are given. Steps to draw Pie-Diagram Step-1: Find aggregate of all components and then by using the following formula

items are Sectorial angles for house rent  1800    360   54  12000 

Sectorial angles for bike maintenance  2400



 360   72 =  12000  Sectorial angles for savings  3000    360   90 12000  

Sectorial angles for miscellaneous  2300    360   64 12000  

Step-2: Now we can draw a circle of suitable radius

C

Step-3: Now we can draw sectors with angles at centre are 75°,54°,72°,90°,69° respectively as shown in the figure. www.betoppers.com

6th Class Mathematics

144

Misc

House rent

Savings

750 690 540 900

Food

Bike Maint 720

Note: While drawing piechart, we can take horizontal radius as base line drawn an angle at the center equal to the degree represented by first component then by taking second line as a base draw an angle equal to the degree of second component and so on till all the components are completed. Line Graph: In a line graph, points are plotted on the graph paper related to two variables. These points are joined in pairs by lines to obtain a linegraph. Linegraph is useful for displaying data or information that changes continuously overtime. Another name for a line graph is a line chart. Line graph has various parts named as tittle, labels, scales, points and lines. They are defined as follows • Tittle : The tittle of the line graph tells us what the graph is about • Labels : The horizontal label across the bottom and the vertical label along the side tells us what kind of facts are listed. • Scales : The horizontal scale across the bottom and the vertical scale along the side tell us how much or how many • Points : The points or dots on the graph show us the facts about given data i.e., the increase or decrease in required components.

Step-4: Plot the points and connect them. Plot a point for each pair of values with items of a pair is indicated by the horizontal scale and by the vertical scale. Then connect the points with straight lines from left to right. Step-5: Finally we can give the graph title. Eg: The number of bicycles manufactured by a company during the year 2006 to 2010 are given in the following table Year

2006

2007

2008

2009

2010

No. of bicycles

1000

800

1100

1500

1400

The line graph for above data as follows: Step-1: Find range in values of years and no. of bicycles. Range of years = 2010 - 2006 = 4 years Range of no.of bicycles = 1500 - 800 = 700 Step-2: Now determine the scales which are suitable and fit for graph along x-axis and y-axis. Scale on x-axis 1cm = 1 units of year; scale on yaxis 1cm = 200 bicycles. Step3: Label the graph along x-axis and y-axis as follows: x-axis as “years”; y-axis as “no.of bicycles” Step-4: Now plot the points for each pair of values of years and no. of bicycles and then connect the points with straight lines. Step-5: Finally given the title for graph as “production of bicycles in a year” The following figure shows line graph Scale on x –1cm = 1 unit y –1cm = 1 unit y - axis

Step for Constructing of Line Graph Step-2: Determine scales. Scales will be depending on greatest values of two sets of components. If we are using graph paper, start with horizontal scale. Let 1cm on the graph paper equal to 1 unit value of given components. While taking scale, we should observer that whether the greatest value will fit on the graph. Continue this process for vertical scale also. Step-3: Label the graph by the units, they represent to mark each unit across horizontal scale and along the vertical scale.

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No. of Bicycles

Step-1: Find the range of two sets of values.

1600 1400 1200 1000 800 600 400 200 2006 2007 2008 2009 2010 Years Production of Bicycles in a Year

x - axis

Data handling

145

Formative Worksheet

(A)

Horror (30%)

Romance (25%)

Computer 1200

18. A survey was conducted among a group of persons of a locality to know the types of books they like to read during leisure time. The given circle graph shows the result of the survey.

English 600 Maths 600

510 Hindi

180 1080 520 0 Hindi 56

Comic (5%)

Russian 30%

Social science

(C) 1080 540

630 540 Maths

(D) 1100

If 99 students opted for Spanish language, then what is the number of students who opted for German language? (A) 159 (B) 198 (C) 266 (D) 363 The given table lists the marks obtained by a student in Maths, Science, English, Computer, Hindi, and Social Science in annual examination. Subject

Science

180

Social science

600

Maths Science English Computer Hindi Social Science

630

Hindi

Spanish 15%

German 55%

20.

Maths 630

630 Science

Science

Social Science

(B)

Autobiography (40%)

If 114 persons like to read horror books during the leisure time, then how many persons were conducted in the survey? (A) 330 (B) 360 (C) 380 (D) 390 19. In a particular school, one of the three languages, Russian, Spanish, or German, is compulsory for the students of class VII, VIII, and IX as an optional paper. The given pie chart represents the number of students (in percentage) opting different language.

180

510

Marks 91 26 91 156 78 78

If the total marks obtained by the student are 520, then which pie-chart correctly shows the given information?

Hindi

600

500

200

Science

600 Maths

Social science

21. A survey was conducted among a group of girls to know their preferences for the types of slippers. The given circle graph shows the results of the survey. Leather Rubber 1400 550 1200

450 Wood

Plastic

If the survey was conducted among a group of 1080 girls, then find out how many girls preferred to wear leather slippers? (A) 140 (B) 280 (C) 420 (D) 560 www.betoppers.com

6th Class Mathematics

146 22. A survey is conducted in a school to know the modes of transport used by students while commuting. The findings of the survey are represented by the given circle graph.

Cycle 800 700 School bus

Walking 1000

500 600 Van Two wheeler

If there are 720 students in the school, then what is the difference between the number of students who come by the most preferred mode of transport and by the least preferred mode of transport? (A) 50

(B) 60

(C) 100

20. A survey was conducted to know the favourite sport of students in a school. The table below gives the data collected from 180 students. Sport

Number of students

Cricket

70

Football

50

Hockey

20

Volleyball

25

Basketball

15

Which pie chart correctly represents the above data? (A) 1600 Cricket

(D) 120 Football 800

Conceptive Worksheet 18. A survey was conducted among 200 students in a school to know the sports they like to play in the evening among football, cricket, hockey, badminton. The given circle graph shows the result of survey conducted. Hockey 30%

(B) 1600 Cricket Football 800

20% Cricket Badminton 25%

Football 25%

(C) How many students like to play badminton in the evening according to the given information? (A) 40 (B) 50 (C) 70 (D) 80 19. The given pie-chart shows the different games liked by the students of a school.

Football 1260 Cricket 900 540

270 630 Volleybal Basketball

Hockey

If 108 students like to play volleyball, then how many students like to play cricket? (A) 144 (B) 180 (C) 252 (D) 360

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Cricket 1400

(D)

Data handling

147 th

21. The given circle depicts the performance of classX students of a school in the board examinations.

2.

3.

1st division 1750 2nd 750 division 500 600 4th division

4. The given pictograph shows the number of flowers sold by a florist on a particular day.

3rd division

Flower

If students getting more than 4 th division are considered pass, then what percentage of students passed the board examinations? (A) 90.25% (B) 83.33% (C) 74% (D) 69% 22. A birthday cake was distributed among 4 friends. The percentage distribution is shown in the form of the adjacent pie chart.

Lisa

10%

30%

Number of flow ers

Rose Tulip Orchid Lily Key : Each

represents 9 flowers

Which flower was sold the most that day? (A) Orchid (B) Tulip (C) Rose (D) Lily

Max 25%

5. Michell

35%

Julis

A group of children carried out a survey about different types and brands o f sh oes. The data was collected and written in the form of a table. Each student in the table represents ten students. Type of shoes

Who got the maximum part of the cake? (A) Julis (B) Michell (C) Lisa (D) Max

Ballerina shoes

The ages of 10 students are 21, 24, 25, 22, 21, 24, 25, 21, 22, and 21. The correct representation of this information is: (A) Age Number

Sn eakers

(B) Age Number

21

3

21

2

22

2

22

2

23

1

23

2

24

2

24

2

25

2

25

2

(C) Age Number

Number of children wearing them

Lace-up shoes

Summative Worksheet 1.

The runs scored by 5 batsmen in a match are 100, 95, 73, x, and 80. If the average runs scored by the batsmen are 81.6, then what is the value of x? (A) 45 (B) 50 (C) 55 (D) 60 The mean of six numbers is 10. Which number should be added to the six numbers so that the mean becomes 11? (A) 16 (B) 17 (C) 67 (D) 77

Keds

Which type of shoes is worn by more than 40 children? (A) Lace-up shoes (B) Ballerina shoes (C) Sneakers (D) Keds

(D) Age Number

21

2

21

4

22

2

22

2

23

1

23

0

24

3

24

2

25

2

25

2

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6th Class Mathematics

148 6.

The bar graph in the given figure shows the favorite activities of the students in a class.

(A) 40% Food

Number of students

y 10  9 8 7 6 5 4 3 2 1

26% Cloth 14% Rent

(B) Sports

7.

Dance Painting Drama Favorite Activity

40% Food

x

Which is the most popular activity among the students in the class? (A) Sports (B) Dance (C) Painting (D) Drama The bar graph in the given figure shows the favorite sports of the students of grade 3 of a particular school.

26% Rent 14% Others

(C) 40% Food

Number of students

y

26% Cloth

45  40  35  30  25  20  15  10  5

14%

(D) 40% Food Badminton

Tennis

Soccer

Baseball

x

26% Others

Favorite sport

As seen in the graph, baseball is the favorite sport of how many 3rd graders? (A) 20 (B) 25 (C) 30 (D) 35 8.

The given table shows the classification of Lucy’s monthly expenditure: Category

Percentage of expenditure

Food

40

Cloth

26

Rent

14

Travelling

10

Others

10

Which of the following pie charts correctly shows the given information?

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14% Cloth

9.

The following pie chart represents the findings of the survey on 1500 people regarding their favourite pastime. How many people like painting? Net surfing 22% Chatting 20% Watching movies 40%

(A) 120 (B) 130

(C) 180

(D) 200

Data handling

149 4.

HOTS Worksheet 1.

The marks of 25 students in M athematics are given in the table below. 15

24

17

21

26

24

15

18

21

28

24

24

18

19

25

17

18

21

19

22

16

24

15

23

14

5.

6.

7.

The tally marks can be represented as (A) (B) (C) (D) 2.

3.

15

17

18

19

21

22

24

25

//

//

//

//

///

/

//////

/

14

15

17

18

19

21

22

/

//

//

//

//

///

/

14

15

16

17

18

19

21

/

///

/

//

///

//

///

15 16 17

18 19 20

//

///

/

//

//

//

/

The given pictograph shows the number of videos rented in four months from a video shop.

2 6 28 /

/

24

25

26

28

////

/

/

/

22

23

24

25

26

28

/

/

////

/

/

/

24

25 26 27

21 22 23 //

/

////

/

Months

/

/

Bart drove through six Midwestern states on his summer vacation. The price of gasoline varied from state to state. The given data shows the gasoline prices in different states: $1.79, $1.61, $1.96, $2.09, $1.84, $1.75 The range of the gasoline prices is (A) $0.75 (B) $0.64 (C) $0.48 (D) $0.31 Age

Number of people

The mean of all the factors of 36 is A and the mean of all the multiples of 2 less than 10 is B. What is the value of A + B? (A) 10.11 (B) 11.11 (C) 15.11 (D) 16.11 The mean of 12 observations is 25. If observations 13 and 21 are removed, then what is the mean of the remaining observations? (A) 20.98 (B) 21.17 (C) 23.2 (D) 26.6 What is the median of even number between the numbers 9 and 23? (A) 14 (B) 16 (C) 18 (D) 20

Number of people who play

group who go to work 8. 10-13 / //// //// //// // 13-16 // //// //// //// //// /// 16-19 //// / //// //// / 19-21 //// //// / //// 22-24 //// //// //// /// 25-28 //// //// //// //// //// // Which of the following statements does not always hold true for the given data? (A) The number of people who go to work increases with an increase in age (B) The number of people who play decreases with an increase in age (C) The number of people who play follows an increasing order relationship with the number of people who go to work for all age-groups (D) The information can be plotted on a double bar graph

Videos rend in the last 4 months

January April July October

Key

= 20 videos

The number of videos rented in the month of October are (A) 20 (B) 40 (C) 60 (D) 80 The number of bananas eaten by a monkey on four days is given in the form of a pictograph: Monday Tuesday Wednesday Thursday

If, it is known that the monkey ate more than two bananas on one of the four days, then the probability that it was a Monday is (A)

1 4

(B)

1 2

(C) 1

(D) 0

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6th Class Mathematics

150 9.

The bar graph in the given figure shows the favorite activities of the students in a class.

35% lottery

(A)

Number of students

y 10  9 8 7 6 5 4 3 2 1

7% shoes

9% clothes 18% savings

60% House rent Sports

Dance Painting Drama Favorite Activity

(B)

x

The total number of student in the class is (A) 20 (B) 22 (C) 24 (D) 26 10. The bar graph in the given figure shows the favorite sports of the students of grade 3 of a particular school.

23% clothes

7% shoes

7% lottery 14% savings

Number of students

y 45  40  35  30  25  20  15  10  5

47% house rent

(C)

39% lottery

5% clothes 13% savings

Badminton

Tennis

Soccer

Baseball

3% shoes

x

Favorite sport

70% house rent

The difference between the numbers of students whose favorite sports are soccer and tennis is (A) 10 (B) 20 (C) 30 (D) 40 11. A local library has 300 books. The adjacent piechart shows the different types of books stocked at the library.

(D)

12% lottery

14% clothes

23% savings

4% shoes 55% house rent

IIT JEE Worksheet

Newsletter

The number of newsletters stocked in the library is (A) 40 (B) 60 (C) 120 (D) 140 12. An individual spends an average of $1,500 on lottery tickets; $5,000 on clothes, $2,000 on watching entertainment shows and $10,000 on house rent in a year. The information may be represented as

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I.

Single Correct Answers

1.

The marks obtained by 10 students in a test are: 45,12,75,55,8,85,95,95,2,93. The range is ? (A) 48 (B) 93 (C) 95 (D) 97 While drawing bargraph, the line vertically is ? (A) x-axis (B) y-axis (C) Orgin (D) xy-axis

2.

Data handling

151

3.

4.

5.

6.

7.

reprersents 30

In which village, was the number of earthquake

victims less than 25%? represents? (A) Village ‘A’ (B) Village ‘B’ (A) 8 (B) 10 (C) 80 (D) 240 (C) Village ‘C’ (D) Village ‘D’ In a bargraph, the items of the data are represented 10. A survey is done on 100 people by ? regarding their preferred genre of (A) Triangles (B) Squares movies. The given table lists the (C) Circles (D) Rectangles results: The scale is 1cm = 250 population in a bargraph, Genre of Percentage of the length of the rectangle which represents 2000 movies people population is ? Science Fiction 30% (A) 500000 (B) 25 (C) 80 (D) 8 Horror 15% In the frequency distribution with classes 1-10,1120,... the upper boundary of the class 1-10 is ? Drama 15% (A) 10.5 (B) 20 (C) 9 (D) 15.5 Action 20% In a bargraph__________all the rectangles is same Comedy 20% ? (A) length (B) Breadth Which of the following pie charts is incorrect? (C) Area (D) Perimeter (A) (B)

II. Multicorrect Answers 8.

Action

The given pictograph shows the number of oranges purchased by a group of four students.

Drama

Comedy

Science friction

Drama Horror

Student Number of oranges

Science friction

(C)

B en

Horror

Comedy

(D) Action

Tim Comedy Horror

Pam

Action

Horror

Drama

Drama

Science friction

Science friction Comedy

Action

11. John

Key: Each

represents 3 oranges

Who purchased the greater than 15 oranges? (A) Pam 9.

(B) Ben (C) Tim (D) John

The adjacent pie chart shows the percentage of people, of different villages, who died during an earthquake. The total number of people who died were 3000.

The given table gives the number of four types of animals in a particular zoo. Animal

Number

Flamingo

4

Panda

7

K oala

5

Kangaroo

6

Which of the following graphs incorrectly represents the given set of data?

Village ‘D’ 15%

20%

35% 30%

Village ‘C’

Village ‘A’

Village ‘B’

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6th Class Mathematics

152 (A) 7  6 5 4 3 2 1 Flamingo Pandu Koala Kangaroo

(B) 7 

13. Three positive integers x, y, and z are such that x + y = 3, y + x = 5, and z + x = 4. What is the mean of x, y, and z? 14. The scale 1cm = 250 Population in a bargraph. The length of the rectangle which represents 1750 population is _____ cm. 15. The marks obtained by 5 students in a test are 92, 94, 96, 98, 100 The range is? 16. represents 5 then represents?

IV. Paragraph

6 4 3 2 1 Flamingo Pandu Koala Kangaroo

(C)

Pandu  Kangaroo  Koala  Flamingo 

(D)

1

2

3

4

5

6

7

1

2

3

4

5

6

7

Pandu  Kangaroo  Koala  Flamingo 

III. Integer Type 12. The number of bananas eaten by a monkey on four days is given in the form of a pictograph: Monday Tuesday Wednesday Thursday The monkey ate a total of 12 bananas over the given period.

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Number of Killed

5

170 165 160 155 150 140 135 130

1

8

15 July

22

29

5

12

July & August

17. Between which two consecutive weeks was the raise in the number of deaths greatest ? (A) 1July - 8July (B) 15July - 22July (C) 22July - 29July (D)5August - 12August 18. In how many weeks more than 145 deaths were registered ? (A) 1 (B) 2 (C) 4 (D) 7 19. In how many weeks less than 165 deaths occurred? (A) 2 (B) 3 (C) 5 (D) 7 20. Between two consecutive weeks was the fall in the number of deaths greatest ? (A) 29July - 5August (B) 5August - 12August (C) 1July - 8July (D) 8July - 15July 21. What is the difference between maximum number of deaths to minimum number of deaths ? (A) 28 (B) 30 (C) 35 (D) 305

V. Matrix Matching (Match the following) 22.

Col-I Col-II (A) {78,76,35,44,60,74,37,65,79,41}, Range is ( ) (B) {10.5, 20.5, 30.5, 40.5, 50.5, 60.5}, Range is ( ) (C) {10110,10111,11001,10001, 11101,10011}, Range is ( )

(p) 37 (q) 40 (r) 44 (s) 1100 (t) 10110

IIT FOUNDATION Class VI

MATHEMATICS SOLUTIONS

© USN Edutech Private Limited The moral rights of the author’s have been asserted. This Workbook is for personal and non-commercial use only and must not be sold, lent, hired or given to anyone else.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of USN Edutech Private Limited. Any breach will entail legal action and prosecution without further notice.

Utmost care and attention to the details is taken while editing and printing this book. However, USN Edutech Private Limited and the Publisher do not take any legal responsibility for any errors or misrepresentations that might have crept in.

Published by

:

USN Eductech Private Limited Hyderabad, India.

CONTENTS 1.

Number System – I

..........

153 - 164

2.

Number System – II

..........

165 - 180

3.

Ratio and Proportion

..........

181 - 190

4.

Algebra

..........

191 - 196

5.

Geometry

..........

197 - 216

6.

Mensuration

..........

217 - 232

7.

Data Handling

..........

233 - 238

1. NUMBER SYSTEM - I SOLUTIONS

FORMATIVE WORKSHEET 1.

We write the given numbers in columns, aligning the ones column 9 2 7 9 0 5 4 8 0  Greatest

7

4.

 

(A)

7

3 < 5, so, 7379 < 7501

5 0 1

3 7 9 1 0

93690562 92791023 92790568

2.

3 7 9

(B)   

1 < 9, so, 37198 < 37910

3 7 1 9 8

Thus, descending order of the given numbers is 927905480, 93690562, 92791023, 92790568 Both the numbers have four digits. 1 2 6 8

4 1

7 3 9 3

(C)     4 1

3 < 5, so, 417393 < 417501

7 5 0 1

 3.

(A)

1 2 8 6 Since 8 > 6, so 1286 > 1268. Rule – 1: The number having more digits is always greter than the number having fewer digits. Rule – 2: If two numbers contain the same number of digits, we compare them with the help of the digits on the extreme left. If thses digits are the same in both the numbers, we compare the digits occupying the next place to its right and so on. 4387 has four digits and 903 has three digits.

7

7

5.

(D)

(E)

(F)

6

3

1

7

0 < 3, so, 747609 < 747631

6

0

9

We write the given numbers in columns, aligning the ones column

6

8

9

5

2

3

7

8

7

2

4

5

8

7

1

3

7

5

2

smallest which is smaller ? greatest

Thus, ascending order of given numbers is: 6895, 23787, 24587, 413752 (b) We write the given numbers in columns, aligning the ones column



(C)

(a)

4

4

So, by Rule 1 4387 is greater than 903, or 4387 > 903. 42183 has the five digits and 7319 has four digits. So, by Rule 1 42183 is greater than 7319, or 42183 > 7319. 80375 has the five digits and 154562 has six digits. So, 154562 is greater than 80375, or 154562 > 80375. Both the numbers 5483 and 6109 have four digits. So, We compare their digits starting from extreme left. Since 6 > 5, so, 6109 > 5483. Both the numbers 51037 and 48876 are of five digits. So, we compare their digits starting from extreme left. Since 5 > 4, so, 51037 > 48876. Both the numbers 293179 and 380113 are of six digits. So, we compare their digits starting from extreme left. Since 3 > 2, so, 380113 > 293179.

7

(D)    



(B)

4

2 2

6.

7.

2

1

7

0

5

1

7

1

5

6

9

3

1

7

2

8

7

1

9

1

smallest which is smaller ? greatest

Thus, descending order of given numbers is: 376319, 374534, 92187, 39124 Smallest 4 digit number is 1000, Smallest 4 digit number having 3 different digits is 1002. Rule: To form the greatest number with the given digits (without repeating any of the digits), we write the greatest digit in the extreme left place; next smaller digit in the next lower place; next smaller digit in the next lower place and so on. In this manner, the smallest given digit is in ones place.

6th Class Mathematics

154

8. 9.

10.

Given digits are 5, 7, 0, 2, 9 Greatest number without repetation 9 7 5 2 0 Rule: To form the smallest number with the given digits (without repreating any of the digits), we write the smallest digit in the extreme left place; next greater digit in the next lower place; next greater digit in the next lower place and so on. In this manner, the greatest given digit is in ones place. Smallest number without repetation 2 0 5 7 9 Greatest 4 digit number is 9 9 9 9, Smallest 4 digit number is 1 0 0 0 Given digits are 9 3 8 1 Greatest 4 digit number from the given digits: 9831 but, we want greatest 4 digit number, 1 is at the third place i.e., greatest four digit number is 9 1 8 3 Smallest 4 digit number from the given digits is 1389 but, we want smallest 4 digit number, 1 is at the third place i.e., smallest four digit number is 3 1 8 9 We write the given numbers in coloumns, aligning the ones column.

3 3

11.

12.

3

9

1

2

4

7

6

3

1

9

9

2

1

8

7

7

4

5

3

4

14.

15. 16.

17.

smallest greatest

18.

which is greater?

Thus, descending order of given numbers is: 376319, 374534, 92187, 39124 Given nuumbers are 432079, 5601729 and 1794805 We read these numbers as Four lakh thirty two thousand seventy nine. Fifty six lakh one thousand seven hundred twenty nine Seventeen lakh nintyfour thousand eight hundred five. We write the given numbers in coloumns, aligning the ones column. 4

3

2

0

7

9

smallest

5

6

0

1

7

2

9

greatest

1

7

9

4

8

0

5

Thus asending order of the following numbers: 432079, 1794805, 5601729 Desending order of the following numbers: 5601729, 1794805, 432079 Given digits are 1, 2, 5, 0 Seven digit numbres from the given digits are: 1025102 5210521 2150125 1252150

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13.

The place value of 4 in thousands place = 4 × 1000 = 4000 The place value of 4 in units place = 4 × 1 = 4 The difference of place values = 4000 - 4 = 3996 Since the face value of a digit in a number is the value of a digit itself, where ever it is, the face value of 4 in thousands place is 4 and the face value of 4 in ten’s place is also 4. Therefore the difference between the face values of two 4’s in 24348 = 4 - 4 = 0 (a) The place value of 8 is thousands. (b) The value of the digit 3 is 30,000. The largest 3-digit number is 999 The largest 2-digit number is 99 The total number of 3-digit numbers = (largest 3-digit number)-(largest 2-digit number) = 999 - 99 = 900 Given number is 39784012 In Indian system we write this number as 3, 97, 84, 012, and read it as three crores, ninty seven lakh, eight foure thousand twelve. In International system we write this number as 39, 784, 012 and read it as thirty ninne million seven hundred eighty four thousand twelve. Given numbers are: 88500784, 32098175 In Indian system we write this number as 8, 85, 00, 784 and 3, 20, 98, 175 In International system we write as 88, 500, 784 and 32, 098, 175 1 8 2 3 0

19. –

9 1 0 8 9 1 2 2

20. × 3 1 3 1 6

6 2 3 2 5

6 5 0 0 0

Smallest 5 digit number is 10000 So that substract 1650 from 10000 1 0 0 0 0 –

Thus,

21.

1 6 5 0 8 3 5 0

Number System – I Solutions

155 32.

22.

2 5 16) 4 0 0 3 2 8 0 8 0 0

33. 23.

24.

4 6)2 4 2 4 0

1 5 214)3 2 2 1 1 1 1 0 3 2

1.8 4 8 6 4 0 8 7 0 8 6 1 4

34.

Total number of students 3520 Number of boys 1928 Number of girls = Total number of students – Number of boys

 1000 1035 

35.

3520 –

1928 1592

26.

Total number of students 3520 Number of boys 1928 Number of girls = Total number of students – Number of boys

36.

37.

3520 –

1928 1592

27.

28. 29. 30. 31.

Total number of questions = 40 Each question carries 2 marks Azar answered 37 questions correctly So that Azar scores 37 × 2 = 74 Marks (a) 2760 (b) 2800 (c) 3000 (a) 19400 (b) 19000 (c) 20000 (a) 165000 (b) 170000 (c) 200000 (a) 3000 (b) 5100 (c) 10600 (d) 7700 (e) 6000 (f) 9000 (g) 19000 (h) 55000

To find the sum by estimation, we estimate the given numbers to the nearest 100, since the least number is of 3 digits.

 300 278 

1 7 2 0 1 7 1 2 8

25.

Here, each number has the same number of digits, i.e., 4 digits. So, we shall estimate each number to the nearest 1000. Thus, the estimated sum = 3000 + 1000 + 3000 + 3000, because = 10000. Here, the given numbers have different number of digits. The least number is of 3-digits. So, we shall prefer to estimate each number to the nearest 100. Thus, the estimated sum = 23, 100 + 7400 + 9300 + 800, because = 40,600.

38.

39.

 300 and 343  Thus, the sum by estimation = 300 + 1000 + 300 = 1600 Since 1600 is close to 1656, the sum 1656 appears to be sensible. Since all the numbers have the same number of digits, we estimate each number ot the nearest 10,000. Thus, the estimated sum = 10,000 + 30,000 = 40,000 Hence, the trader will be able to make the payment. Here, we shall estimate each number to the nearest 100.  6500; and 431   400 Thus, 6527  Hence, the estimated difference = 6500 – 400 = 6100. Estimated number of workeers when estimated to the nearest 1000 = 12000 Estiamted number of male workers = 7000 Therefore, the estimated number of female workers = 12000 – 7000 = 5000. (a) 87  90 and 313  300. So, estimated product = 90 × 300 = 27000 (b) 958  1000 and 387  400. So, estimated product = 1000 × 400 = 400000 (c) 8193  8000 and 247  200. So, estimated product = 8000 × 200 = 1600000 (A) XLII = (50 – 10) + 2 = 40 + 2 = 42 (B) XLIV = (50 – 10) + (5 – 1) = 40 + 4 = 44 (C) LXVI = (50 + 10) + 6 = 60 + 6 = 66 (D) XCVIII = (100 – 10) + 8 = 90 + 8 = 98

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6th Class Mathematics

156 40.

47.

(A) 58 = 50 + 8 = LVIII (B) 42 = (50 – 10) + 2 = XLII (C) 97 = (100 – 10) + 7 = XCVII (D) 66 = (50 + 10) + = LXVI.

41.

M = 1000, C = 100, X = 10, I = 1, V = 5

1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 20 .

48.

(E) Associativity for multiplication 49.

51.

28 – 0

2

43.

14

–0

7

–0

2

3

–1

1

–1

(A) IXX (D) LIL

(B) IVC

(F) LIV

(H) VVII

45.

(A) LXXI, LXXII, LXXIII, LXIV (B) XCI, XCII, XCIII, XCIV

46.

XCVIII, XCI, LXIII, XL

59.

a) b) c)

d) e)

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(A) 173

(B) 551

(C) 256

(D) 823

(E) 457

(F) 245

(A) 3600

(B) 18600

(C) 55000

(D) 17000

(A) 383

(B) 751

(C) 5300

(D) 18600

52.

Two distances are equal

53.

 v ,B   i ,C   iii ,D   iv A   ii , ,E 

= 111000(2) 44.

(A) Commutatitity for addition

(D) Associativity for addition

56

2

(D) CCCLXXX

(C) Identity for multiplication

50.

2

(C) CML

(B) Identity for addition

32 + 16 + 0 + 0 + 2 + 1 = 32 + 19 = 51.

2

(B) CDLXXX

(E) MMCDXIII

The rule that the roman numeral after the greastest roman numeral is added and before the greatest roman numaral is substracted 1000 + 100 + 10 + 5 – 1 = 1114 42.

(A) DCCCX

(C) IC

54.

4171 km

55.

Rs. 1344600

56.

Rs. 156686

57.

(a) Decrease in weight south

(b) 30 km

(c) 326 A.D. (d) Gain of Rs. 700 (e) 100 m below sea level. 58.

(a) + 2000

(b) – 800

(c) + 200

(d) –700.

Number System – I Solutions

157

60.

61.

(a)

–10°C, –2°C, + 30°C, + 20°C, –5°C

(b)

62. 63. 64. 65. 66.

(c) Siachin (d) Ahmedabad and Delhi. (a) 9 (b) –3 (c) 0 (d) 10 (e) 6 (f) 1 (a) –6, –5, –4, –3, –2, –1 (b) –3, –2, –1, 0, 1, 2, 3 (c) –14, –13, –2, –11, –10, –9 (d) –29, –28, –27, –26, –25, –24. (A) –19, –18, –17, –16 (B) –11, –12, –13, –14. (a) T (b) F; –100 is to the left of –10 on a number line. (c) F; greatest negative integer is –1. (d) F; –26 is smaller than –25. (a) We want to get integer on moving 4 numbers to the right of –2. So we begin from –2 and proceed 4 units to the right of –2 as shown in fieure.

(b)

67.

(c) (d) (a)

Hence we well reach the number 2. We want to obtain integer on moving 5 numbers. So we begin from 1 and proceed 5 units to the left of 1 to obtain –4 as shown in the figure.

Hence, we well reach the number –4. Since –13 < –8. So to reach –13 from –8 on the number line we should move to left of –8. Since –1 > –6 So to reach –1 from –6 on the number line, we should move to right of - 6. Start from number 4, move 5 units to the positive direction (+5). Move 5 units to the right.

4

5

6

7

8

9

10

 4 + (+5) = 4 + 5 =9

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6th Class Mathematics

158 (b)

Start from number 3, move 4 units to the negative direction (–4).

75.

(a) Start from 7, move 6 units to the left.

Move 4 units to the left.

2 1

0

1

2

3

4

0

 3 + (–4) = 3 – 4

= –1

1

2

3

4

5

6

7

 7 – (+6) = 7 – 6

Move 3 units to the left.

(b)

=1 Start from –8, move 3 units to the right.

68. 9

7 6

8

5

4

Start from 8, move 3 units to the right.

 –5 + (–3) = –5 – 3

69.

70.

71.

72.

=–8 – 3 + 7 + (–8) =–3+7–8 =4–8 =–4 The distance of the pendulum from the table = – 80 + 35 = – 45 cm  The pendulum is 45 cm below the table. The question involves the sum of negative integer and a positive integer. –14 oC + 7 oC = – 7 oC The temperature of the town during the day is –7 oC. (a) 11 + ( – 7) = 11–7 = 4 (b) – 13 + ( + 18)= –13 + 18 = 5 (c) (d) (e)

73.

74.

– 10 + ( + 19) = –10+ 19 – 9 (–250) + (+ 150) = – 250 + 150 = – 100 ( – 380) + ( – 270) = – 380 – 270

= –650 (/) ( – 217) + ( – 100) = –217– 100 = –317. (a) 137 and –354 = 137 + (–354) = 137 – 354 = –217 (b) – 52 and 52 = – 52 + 52 = 0 (c) –312, 39 and 192 = 39 + 192+ (–312) = 231–312 – 81 . (d) – 50, – 200 and 300 = 300 + ( – 200) + ( – 50) = 300 + ( – 200 – 50) = 300 – 250 = 50. First, find the pattern of the number sequence. Then, find the value of x and y. Add x and y. 2 divisions x

8

x = – 8 + (–4) the left means (–4)) = – 12 y=–4+6 the right means (+6)) =2  x + y = – 12 + 2 = – 10

5 4

=–5 76.

– 8 – (+3) – (–5) =–8–3+5 = – 11 + 5 =–6

77.

Initial position = – 100 m Final position = – 100 m + (–20 m) + 35 m = – 100 m – 20 m + 35 m = – 85 m  He is 85 m below sea level.

78.

(a)

35 - 20 = 15

(b)

72 - (90) = 72-90 = – 18

(c)

(–15) – (–18) = –15 + 18 3

(d)

(–20) –(13) = –20 – 13 = –33

(e)

23 – ( – 12) = 23 + 12 = 35

(f)

(– 32) – (– 40) = – 32 + 40 = 8

(a)

( – 3) + ( – 6) < ( – 3) – (– 6)

(b)

(–21) – (–10) > (–31) + (–11)

(c)

45 – (–11) > 57 + (–4)

(d)

(–25) – (–42) > (–42) – (–25)

(a)

– 8 + 8 = 0 Ans.

(b)

13 + ( – 13) = 0 Ans.

(c)

12 + (– 12) = 0 Ans.

(d)

( –4) + (–8) = – 12 Ans.

(e)

( + 5) – 15= – 10 Ans.

79.

80.

y

4

8 7 6

 – 8 – (–3) = – 8 + 3

3 divisions

2 divisions = 4  1 division = 2

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9

81.

(a) (– 7) – 8 – ( – 25) = – 7 – 8 + 25 = – 15 + 25 = 10

(Two divisions to

(b) (– 13) + 32 – 8 – 1 = – 13 – 8 – 1 + 32 = – 22 + 32 = 10

(Three divisions to

(c) (– 7) + ( – 8) + ( – 90) = – 7 – 8 – 90 = – 105 (d)

50 – ( – 40) – ( – 2) = 50 + 40 + 2 = 92

Number System – I Solutions

159

CONCEPTIVE WORKSHEET 1. 2. 3.

4. 5.

6.

7.

8.

9.

10.

11. 12.

13.

14.

15.

Greastes 6 digit number is 999999 and its successor is 1000000 Smallest 7 digit number is 1000000 and its predecessor is 999999 (a) 97501 (b) 732156 (c) 100000 (d) 63354 (a) 3921 (b) 2333 (c) 10249 (a) 4370, 4425, 4536, 4928 (b) 25068, 25245, 25270, 25510 (c) 28956, 52896, 68952, 69825 (a) 78501, 10378, 8570, 7508 (b) 52430, 45032, 30254, 23450 (c) 201010, 110021, 102021, 22101 Greatest and smallest 4 digit numbers from the given digits are Greatest Smallest a) 7531 1357 b) 9732 2379 c) 6510 1056 d) 7540 4057 Greatest and smallest 5 digit numbers from the given digits are: Greatest Smallest a) 66662 22226 b) 77731 11137 c) 88642 22468 d) 77530 30057 Smallest 7-digit number: 1000000 smallest 7-digit number having 5 different digits: 1000234 Greatest 6-digit number: 999999 Greatest 6-digit number having 4 different digits: 999876 (a) 300 (b) 30 (c) 3 (d) 3000 (e) 300000 (a) 2 × 1000 + 1 × 100 + 0 × 10 + 3 × 1 (b) 7 × 10000 + 5 × 1000 + 1 × 100 + 3 × 10 + 2×1 (c) 1 × 100000 + 0 × 10000 + 7 × 1000 + 9 × 100 + 1 × 10 + 2 × 1 (d) 3 × 100000 + 1 × 10000 + 7 × 1000 + 9 × 100 + 5 × 10 + 6 × 1 (e) 3 × 1000000 + 7 × 100000 + 5 × 10000 + 4 × 1000 + 0 × 100 + 3 × 10 + 1 × 1 (A) 2,69,00,271 (B) 4,75,29,311 (C) 9,43,81,052 (D) 70,83,56,149 (E) 62,32,79,845 (F) 55,30,088 (A) 1,974,522 (B) 30,706,210 (C) 42,358,761 (D) 86,465,099 (E) 71,531,680 (F) 2,795,814 (a) six crore, eighty-seven thousand, nine hundred seventy-one

16.

17.

18. 19. 20.

21. 22.

23. 24.

(b) thirty-six crore, twenty lakh, eight hundred four (c) fourty crore, fifty-five lakh, one thousand (d) two crore, ninety-five lakh, twenty-five thousand, six hundred sixty-six (e) seven crore, twenty-five lakh, thirty-five thousand, four hundred sixty-nine (f) eighty-three crore, seventeen lakh, twentynine thousand, six hundred fourteen (a) seven million, seventy-one thousand, seventy-one (b) thirty-six million, seven hundred fifty-four thousand, nine hundred eighty-one (c) twenty-three million, sixty-four thousand, five hundred ninety-four (d) ninety million, ninety thousand, ninety (e) two hundred fifty-four million, seven hundred eighty-nine thousand, six hundred ten (f) Six hundred eighty-five million, three hundred twenty-one thousand, four hundred seventy-nine (a) 20,29,00,717 (b) 8,13,40,012 (c) 19,90,14,680 (d) 50,40,60,001 (e) 7,23,86,594 (f) 6,352,946 (g) 49,782,058 (h) 90,000,009 (i) 20,380,100 (j) 81,412,650 (i) 98 (ii) 89 (iii) 40 (iv) 1354 (v) 773 (vi) 88 (i) 2136 (ii) 1000 (iii) 0 (iv) 44 (v) 6367 (vi) 863 (i) 32 + 46 – 12 = 78 – 12 = 66. (ii) 75 + 38 – 15 = 113 – 15 = 98. (iii) 48 – 28 + 76 + 24 = 20 + 76 + 24 = 120 (iv) 46 + 91 – 46 – 91 = 137 – 46 – 91 = 91 – 91 = 0 (v) 754 – 54 + 154 – 254 = 700 + 154 – 254 = 854 – 254 = 600 (i) 100 (ii) 0 (iii) .0 (iv) 2 (v) 56789 (vi) 7 (i) 336  5 = 1680 (ii) 24  25 = 600 (iii) 78  5 = 390 (iv) 7  125 = 875 (v) 65  5 = 13 (vi) 575  25 = 23 (vii) 105  5 = 21 (viii) 325  25 = 13 (i) 252 (ii) 28 (iii) 12 (iv) 1476 (v) 4 Van can carry 12 people Vans required to carry 108 people =

25.

108 9 12

Nlini have 15 tennis balls She gives 7 tennis balls tyo her friend i.e., 15 – 7 = 8 tennis balls. Her father brings back 5 tennis balls to her. Now she is having 8 + 5 = 13 tennis balls.

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6th Class Mathematics

160 26.

Madan brought 20 boxes of pens Each box contain 15 pens. Total pens = 20 × 15 = 300. Pens were equal distributed among 25 students

300  12 25  Each student recive 12 pens. Raghu monthly salary is 2,700 Annual salary of Raghu = 12 × 2700 = 32,400. Total bollons = 36. Each child can get 6 bollons Total number of children can get 6 bollons = i.e.,

27. 28.

29.

30. 31. 32.

33.

37.

36 6 6 Total number of mango trees = 5100 Number of rows = 17 Total number of trees in each row =

38.

5100  300 17 (A) 90,000 (B) 86,000 (C) 86,200 2,49,000 Gives that GenX automobile costs Rs. 27,895 Gen-Y automobile costs Rs. 12,329 27,895 to nearest thousand is = 28000 12,329 to nearest thousand is = 12000 Estimated difference 28000 – 12000 16000  Gen X is 16000 more cast than Gen Y mobile. (i) 630 (ii) 390 (iii) 1100

40.

39.

41.

42.

43. 44. 45. 46. 47.

34.

(i)

30

(ii)

60 48.

40

70

40

20

20

+ 20

50.

130

170

51.

49.

+

(iii)

120 140 + 150 410 35. (a) 7300 (c) 26, 500 36. (A) 50 × 30 (B) 50 × 30 (C) 50 × 80 (D) 40 × 20 (E) 60 × 40 (F) 50 × 90 (G) 400 × 30 www.betoppers.com

52. 53. (b) 73, 900 (d) 4, 100 = 1500 = 1500 = 4000 = 800 = 2400 = 4500 = 12000

54.

55. 56.

(H) 100 × 90 = 9000 (I) 200 × 80 = 16000 (J) 400 × 200 = 80000 (K) 400 × 100 = 40000 (L) 900 × 200 = 180000 (M) 5000 ×100 = 500000 (N) 5000 × 200 = 1000000 (O) 2000 × 100 = 200000 (A) XVII (B) XXIII (C) XLVIII (D) LXXV (E) XCV (F) XIX (G) XLI (H) XCIII (I) LXXXVII (J) LXIV (A) 24 (B) 71 (C) 91 (D) 49 (E) 75 (F) 99 (G) 86 (H) 61 (A) XLII = (50 – 10) + 2 = 40 + 2 = 42 (B) XLIV = (50 – 10) + (5 – 1) = 40 + 4 = 44 (C) LXVI = (50 + 10) + 6 = 60 + 6 = 66 (D) XCVIII = (100 – 10) + 8 = 90 + 8 = 98 (A) 58 = 50 + 8 = LVIII (B) 42 = (50 – 10) + 2 = XLII (C) 97 = (100 – 10) + 7 = XCVII (D) 66 = (50 + 10) + = LXVI. (A) 3400 = MMMCD (B) 1035 = MXXXV (C) 1856 = MDCCCLVI (D) 1872 = MDCCCLXXII (A) XLII < XLIV (B) LVII < LXVI (C) XCVII < C (D) LXXI > XLIX 1,2,3,4....... 1 0 (A) 20 (B) 71 (C) 313 (D) 2020 (E) 13701 (A) 36 (B) 218 (C) 702 (D) 1899 (E) 51375 (A) 317 (B) 933 (C) 7091 (D) 8395 (A) T (B) T (C) T (D) F (E) T (F) F (G) F (H) T (I) F A) 35 (B) 173 (C) 83 (D) 17 (E) 17 (F) 101 (A) F (B) F (C) F (D) T (E) T (F) T (G) T (H) F (A) 0 and 1 ( B) 0 (C)any nutural number (D)12 (A) 7020 (B) 6790 (C) 7500 (D) 6030 (E) 5200 (F) 753000 (G) 9750 (H) 43800 (A) T (B) T (C) F (D) F (E) T (F) T (G) F (H) F (I) T (J) F (K) T (L) T (M) F (A) – 18 0C (B) 18 0C (a) Horizontal number line: 15 10 5

0

5

10 15

Number System – I Solutions (b)

161

Vertical number line: 15 10 5 0 5 10 15

2 lies to the left of 3. Therefore, 2 is less than 3.

57. 3

58. 59.

2

1

1

0

2

3

4

Therefore, –2 is the smaller integer. Therefore, – 3, is greater than – 5. Arrange the integers on a number line. (A) Ascending order: From smallest to largest Values increasing 5

(B)

4

3

2

1

1

0

2

4

3

5

6

7

Integers in increasing order: –5, –2, 0, 2, 3, 6, 7  Descending order: From largest to smallest. Values decreasing 8

60.

7

6

+5

62. 63.

64.

65.

3

2

1

0

1

2

3

4

5

 Integers in descending order : 5, 3, –1, –2, –5, –8 Arrange the integers in ascending order: –12, –10, –9, 0, 5, 7, 8  The largest integer is 8 and the smallest integer is –12. 15, 10, 5 , 0 , 5 , 10,

61.

4

5

+5

+5 +5

+5

15 +5

Difference between consecutive integers = 10 

(A) (B) (C) (A) (B) (C) (D) (A) (B) (C) (D)

(i) +18 m (ii) – 8 m (i) +Rs. 188 (ii) –Rs. 254 (i) +15 km (ii) –30 km (–11) + (–12) = –23 (+10) + (+4) = 14 (–32) + (–25) = –57 (+23) + (+40) = +63 (–7)+ (+8) = (–7) + (+7) + (+l) = 0 + (+l) = +l (–9) + (+13) = (–9) + (+9) + (+4) = 0 + (+4) = +4 (+7) + (–10) = (+7) + (–7) + (–3) = 0 + (–3) = –3 (+12) + (–7) = (+7) + (+5) + (–7) = 0 + (+5) = 5

(A) First we move 2 steps to the left of 0 reaching –2 and then from this point we move 6 steps to right. We reach the point 4. Thus (–2) + 6 = 4.

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6th Class Mathematics

162

(B)

66.

67.

First we move 6 steps to the left of 0 reaching -6 and then from this point we move 2 steps to right. We reach the point -4. Thus (-6) + 2 = -4. (A) (+7) + (-11) = (+7) + (–7) + (-4) = 0 + (–4) = -4 (B) (–13) +(+10) = (–10) + (–3) + (+10) = 0 + (–3) = –3 (C) (–7)+ (+9) = (–7) + (+7) + (+2) = 0 + (+2) = 2 (D) (+10) + (-5) = (+5) + (+5) + (-5) = (+5) + 0 = +5. (A) We want to get an integer 3 more than 5. So we start from 5 and proceed 3 units to the right to obtain 8 as shown in the figure.

(B)

68.

We want to get an integer 5 more than – 5. So we start from – 5 and proceed 5 unit to the right to obtain 0. as shown in figure.

(C)

Hence, 5 more than – 5 is 0. We want to get an integer 6 less than 2. So we begin from 2 and proceed 6 units to left to obtain – 4 as shown in figure.

(D)

Hence 6 less than 2 is - 4. We want to get an integer 3 less than - 2. So we start from - 2 and proceed 3 units to left of it to obtain – 5 as shown in figure.

Hence, 3 less than - 2 is - 5. The left of 9 reaching 3 as shown in the figure.

Thus, 9 + (- 6) = 3 (B) First we move to the right of 0 by 5 steps reaching 5. Then, we move 11 steps to the left of 5 reaching - 6 as shown in the figure.

(C)

Thus, 5 + (- 11) = -6 First we move to the left of 0 by 1 step reaching - 1. Then we move 7 steps to the left of - 1 reaching - 8 as shown in the figure.

Thus, (- 1) + (- 7) = - 8 www.betoppers.com

Number System – I Solutions (D)

69. 70. 71. 72. 73. 74. 75.

163

First we move to the left of 0 by 5 step reaching - 5. Then we move 10 steps to the right of- 5 reaching 5 as shown in the figure.

(E)

Thus, ( - 5) + 10 = 5 First we move to the left of 0 by 1 steps reaching - 1. Then we move 2 steps to the left of - 1 reaching 3. Again, we move 3 steps to the left of -3 reaching - 6 as shown in the figure.

(F)

Thus, (–1) + (-2) + ( -3) = - 6 First we move to the left of 0 by 2 steps reaching - 2. Then we move 8 steps to the right of - 2 reaching 6. Again, we move 4 steps to the left of 6 reaching - 2 as shown in the figure.

Thus, ( - 2) + 8 + ( - 4) = 2 (i) 9 (ii) (–28) (iii) 0 (i), (ii), (iii), (iv). (i) –52 (ii) 40 (iii) –14 (vi) 92 (vii) –115 (viii) 2481 No. –33 1 (i) –3 (ii) 9 (iii) – 4 (iv) 0

(iv) –100

(v) –8

(iv) –105 (ix) –6511

(v) –219 (x) 9262

(v) 6

(vi) –389.

(vi) –91

SUMM ATIVE WORKSHEET KEY

24. 25.

1

2

3

4

5

6

7

8

9

10

11

12

A

D

C

A

D

A

B

D

D

C

D

D

13

14

15

16

17

18

19

20

21

22

23

A

A

B

B

C

D

B

A

D

C

D

(i) 3 (i) – 452 (viii) –246 26. (i) –22 27. (i) 9, 31. –1168

(ii) 4 (ii) 988 (ix) 2114 (ii) –59 (ii) –7 32. 15

(iii) 0 (iv) –3 (iii) –2858 (iv) 991 (x) – 30. (iii) –372 (iv) 276 28. (ii); (vi). 33. 54.

(v) –6 (vi) – 3 (v) –1421. (vi) 64

29. – 587

(vii) – 81

30. 72 m

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6th Class Mathematics

164

HOTS WORKSHEET 1. 6. 8. 9. 12. 16. 18. 19. 20.

11999 2. 2954 3. 290337 4. 25489 5. 13340 6260 7. 13890 (i) 10 (ii) 110 (iii) 648 (iv) 92 (v) 572 20 rolls 10. 4,780 books, Rs. 47,800 11. Rs. 1,568 8760 hrs., 8784 hrs., 24 hrs. 13. –317 14. –282 15. cooler 2 0C 17. –10 –181; 181 ft below the surface – 260; 260 m below the surface –16 21. –5 22. –12 0C 23. 26,849 ft

IIT JEE WORKSHEET Q.no

1

2

3

4

5

6

7

8

9

10

11

12

Ans

D

D

A

A

B

B

B

A

C

C

C

B

Q.no

13

14

15

16

17

18

19

20

21

22

23

24

Ans

D

C

B

B

D

A

C

A

A, B, C

A, B, D

B, C

B, D

Q.no

25

26

27

28

29

30

31

32

33

34

35

36

Ans

A, C

A, B, D

A,D

B, C

B, C

A, C

C

A

B

B

D

D

Q.no

37

38

39

40

41

42

43

44

45

46

47

48

Ans

A

B

C

A

D

D

D

C

B

A

0

6

Q.no

49

50

51

52

53

54

55

56

Ans

5

8

4

0

9

2

A  s , B p, t, C q, D p, t

A  t , B v, C q, D p

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2. NUMBER SYSTEM – II SOLUTIONS

FACTORS & MULTIPLES

= 20, which is divisible by 4.

FORMATIVE WORKSHEET – 1

 75020 is divisible by 4.

1.

Numbers between 25 and 30 are 26, 27, 28, 29.

(ii) Here, the number formed by the last two

Factors of 26 are 1, 2, 13, 26.

digits

Sum of factors = 1+ 2 + 13 + 26 = 42

= 42, which is not divisible by 4.

Which is not equal to two times 26 (i.e. 52)

 987542 is not divisible by 4.

 26 is not a perfect number

4.

(i)

Similarly we can check 27 and 29 are not perfect

 460765 is divisible by 5.

numbers.

(ii) Here, unit’s digit = 2

Factors of 28 are 1, 2, 4, 7, 14, 28

 50502 is not divisible by 5.

Sum of factors = 1 + 2 + 4 + 7 + 14 + 28

(iii) Here, unit’s digit = 0

= 56 = 2  28  28 is a perfect number..

2.

(i)

Here, the sum of digits in the given number 932105 is = 9 + 3+ 2 + 1 + 0 + 5 = 20,

5.

(i)

Here, the unit’s digit = 6  the given number is divisible by 2.

Also, the sum of the digits

 932105 is not divisible by 3.

= 2 + 4 + 0 + 5 + 6 = 17

4980204 is = 4 + 9 + 8 + 0 2 + 0 + 4 = 27, which is divisible by 3.  4980204 is divisible by 3.

(iii) Here, the sum of digits in the given number

(i)

 13790 is divisible by 5.

which is not divisible by 3.

(ii) Here, the sum of digits in the given number

3.

Here, unit’s digit = 5

which is not divisible by 3, so the given number is not divisible by 3. Hence, 24056 is not divisible by 6. (ii) Here, the unit’s digit = 4  the given number is divisible by 2

Also, the sum of the digits

262242 is

= 9 + 8 + 2 + 7 + 4 = 30

= 2 + 6 + 2 + 2 + 4 + 2 = 18,

Which is divisible by 3,

which is divisible by 3.

so the given number is divisible by 3.

 262242 is divisible by 3.

Hence, the given number 98274 is divisible

Here, the number formed by the last two digits

by 6.

6th Class Mathematics

166 6.

(i)

Here, the number formed by the last three

FORMATIVE WORKSHEET – 2

digits = 048 i.e. 48 which is divisible by 8.

1.

180  4 = 45

 987048 is divisible by 8.

180  5 = 36

(ii) Here, the number formed by the last three digits = 842, which is not divisible by 8.

 180 is a common multiple of 3, 4 and 5.

2.

 5719842 is not divisible by 8.

7.

(i)

The factors of 6: 1,2,3,6 The factors of 8: 1,2,4,8

Here, the sum of digits

The factors of 12: 1,2,3,4,6,12

= 6 + 3 + 4 + 6 + 8 + 0 = 27,

 The common factors of 6, 8 and 12 are 1

which is divisible by 9.  634680 is divisible by 9.

180  3 = 60

and 2. 3.

10 = 1  10 = 2 5

(ii) Here, the sum of digits

Factors of 10 are {1, 2, 5, 10}

= 4 + 2 + 0 + 4 + 5 + 6 + 1 = 22, which is not divisible by 9.

15 = 1  15 = 3 5

 4204561 is not divisible by 9.

8.

(i)

Here, unit’s digit = 5

Factors of 15 are {1, 3, 5, 15}.

 500505 is not divisible by 10.

The common factors of 10 and 15 are { 1, 5}

(ii) Here, unit’s digit = 0  8179320 is divisible by 10.

9.

(i)

4.

(i) 2 90

(90 is divisible by prime 2)

3 45

(45 is divisible by prime 3)

Here, the sums of the digits at the alternate

3 15

(15 is divisible by prime 3)

places are 2 + 5 + 2 + 2 and 8 + 1 + 2 i.e.

5

11 and 11 Their difference = 11 – 11 = 0,



(ii)

(5 is itself prime)

90 = 2  3  3  5 3 675

(675 is divisible by prime 3)

which is divisible by 11.

3 225

(225 is divisible by prime 3)

 2221582 is divisible by 11.

3 75

(75 is divisible by prime 3)

5 25

(25 is divisible by prime 5)

(ii) Here, the sums of the digits at the alternate

5

(5 is itself prime)

places are 9 + 4 + 5 + 2 and 3 + 5 + 4 i.e.20 and 12



Theire difference = 20 – 12 = 8, which is

(iii)

not divisible by 11.  the given number 2455439 is not

3 1089

(1089 is divisible by prime 2)

3 363

(363 is divisible by prime 3)

11 121

(121 is divisible by prime 11)

11

divisible by 11. 

www.betoppers.com

675 = 3  3  3  5  5

(11 is itself prime)

1089 = 3  3  11 1  11. 1.

Number System – II Solutions 5.

167

Set of factors of 72 = F(72)

9.

= {1, 2, 3, 4, 6, 12, 18, 24, 36, 72}

6  2 3

9 = 3 3

We find that 2 occurs once where as 3 occurs a prime factor maximum 2 times.

Set of factors of 192 = F(192)



= {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192}

L.C.M. = 2  3  3 = 18.

10. We know H.C.F.  L.C.M. = Product of two

Set of factors of 324 = F(324)

numbers = {1, 2, 3, 4, 6, 9, 12, 27, 36, 54, 81, 108, 162, 324}

4  L.C.M. = 336 336  84 i.e. L.C.M. = 4 

Set of common factors of 72, 192 and 324 = F(72)  F(192)  F(324) = {1, 2, 3, 4, 6, 12} 11. The highest element of this set is 12.  H.C.F. of 72, 192, 324 = 12 Common factor

2 3

6.

6, 12 3, 6 1, 2

LCM = 3  2  3  20 = 360

The division stops here because 1 and 2 do not have any common factors except 1.

HCF = 2  3 (Multiply the common factors)

8.

3 and 20 are co-prime numbers. So their product LCM.

2 36 2 18 3 9 3 3 1

36 = 2  2  3

24 and 60 are factors of 120. All multiples of 120 will naturally be multiples 24 and 60. So 24 and 60 are struck out.

is taken directly and that product will be the

=6 7.

3 18, 24, 60, 120 2 6, 40 3, 20

2 48 2 24 2 12 2 6 3 3 1

12. 120

136

1

120 16

120 7 112

;

8

16 2

348 = 2  2  2  2  3

16

On circling the common factors, we obtain the GCD = 2  2  3 = 12

0

The remainder is 0. The last divisor is 8. The

2 6, 9

GCD of the two numbers is 8.

2, 3 Note: If the last divisor is 1, the GCD is1, and (3 is a common factor of 6 and 9) (2, 3 have no common factor)

the number are co-prime. 13. The GCD of any two of the three numbers is

 L.C.M. = 3  2  3 = 18.

first obtained using the division method.

B. Prime factorisation method

Consider 459 and 357

In prime factorisation method, we write the

357

459 1

prime factorisation of each of the given

357

numbers. L.C.M. is equal to the product of all

102

the different prime factors of the given numbers

357 3 306

using each common prime factor, the greatest

51 102 2

number of time it appears in the prime

102

factorisation of any of the given numbers.

0

51 is the GCD of 357 and 459. www.betoppers.com

6th Class Mathematics

168 The GCD of the third number 306 and the result

3.

2

136

of the first division i.e. 51 gives us the GCD of

2

68

the three numbers.

2

34

2

17

Next we find the GCD of 51 and 306 Therefore 136 =23

51 306  6

171

306

The number of divisors (factors) of given

0

number n = apbq is (p + 1) (q +1) where a, b are primes.

The GCD of 459, 357 and 306 is 51. Therefore number of positive divisors of

CONCEPTIVE WORKSHEET -1 1.

(i) 24

(ii) 11

(iii) 29

(iv) 18 ;

(a) even (b) odd

(c) odd

(d) even

(i) 259

(ii) 252

(iii) 414

(iv) 208

(a) odd

(b) even (c) even

136 = (3 + 1) (1 + 1) = 8 4.

Since sum of all digits in a given number 156 = 1 + 5 + 6 = 12 which is divisible by 3 and 156 = 13 12

2.

Hence 156 is divisible by 3, 13 Therefore 156 is common multiple of both 3, 13 3.

D

4.

C

5.

By definition of even numbers, the set of all

5.

Given sum of all the prime factors of a number = 15 Now 15 can be written as sum of two prime

whole numbers divisible by 2 are called even

numbers = 2 + 13 = 15

numbers. Product of prime factors = 2

13 = 26

Here - 2 is negative integer and 3 is odd natural In given options, only 104 is divisible by 26.

number

Therefore the required number = 104

Therefore 2 is even number 6.

CONCEPTIVE WORKSHEET -2 1.

Since 48 = 2

twin primes. 3, 5 are twin primes. 7.

24

Pair of primes whose difference is 2 are called

To determine whether 4 is a common factor, carry out the division method.

120 = 5

24

192 = 8

24

36  4 = 9 44  4 = 11 1 48, 120, 192 are multiples of 24 100  4 = 25 2. 36, 44 and 100 are all divisible by 4. 2

96

2

48

2

24

2

12

2

6

 4 is a common factor of 36, 44 and 100.

8.

= 2  10

3

Therefore 96 = 25

= 4 5 3

Therefore the prime factors of 96 are 2, 3 www.betoppers.com

20 = 1  20

The factors of 20 are {1, 2, 4, 5, 10, 20}

Number System – II Solutions 9.

169

The factors of 14: 1,2,7,14

30 = 2  3  5; 75 = 3  5  5 ;

The factors of 20: 1,2,4,5,10,20

135 = 3  3  3  5

The common factors of 14 and 20 is 1, 2.

The GCD 3  5 = 15

 The HCF of 14 and 20 is 2.

SUMM ATIVE WORKSHEET

(Choose the largest common factor). 10. (i)

Set of factors of 16 = F(16)

1.

{1, 2, 3, 4, 6, 8, 12, 16, 24, 48}

= {1, 2, 4, 8, 16}

{1, 2, 3, 4, 5, 10, 20, 25, 50, 100} ;

Set of factors of 24 = F(24)

{1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144}

= {1, 2, ,3, 4, 6, 8, 12, 24} Set of common factors of 16 and 24 = F(16)  F(24) = {1, 2, 4, 8}

2.

{1, 5}; {1, 3, 9} {1, 2, 4, 8, 16}

3.

(i) 2  3 2  7

(ii) 5  3  2 4

(iii) 2 4  3 2

(iv) 5  7  3 2

The greatest or largest (or highest) element

(v) 2 2  3 2  13

of this set is 8. 

H.C.F. of 16 and 24 = 8

4.

(v) 23  3  11 1  13

= {6, 12, 18, 24, 30, 36, .....} Also the set of multiples of 9

5.

= M(9) = {9, 18, 27, 36, ....} Set of common multiples of 6 and 9

6.

= M(6)  M(9) = {18, 36, .....} The smallest element of this set is 18. L.C.M. of 6 and 9 = 18.

12. Given numbers are 24 and 40. 24 = 2  2  2  3 

H.C.F.  L.C.M. = 8  120 = 960 .... (1)

(iv) 300

(v) 18

(i) 50

(ii) 158

(iii) 1

(iv) 11

8.

12

9.

(i) 1890

(ii) 180

(iv) 1170

(v) 38220

10. (i) 960 (iv) 21780

(iii) 36

(v) 6

(iii) 4704

(ii) 3600 (iii) 1800 (v) 74750

11. 6

Also product of the two numbers

12. 7

= 24  40 = 960.... (2)

13. {1, 3, 9, 27} ; {1, 2, 4, 13, 26, 52}

From (1) and (2), we get the required result. 13.

(ii) 53

36

L.C.M. = 2  2  2  3  5 = 120 

(i) 23

7.

40 = 2  2  2  5

H.C.F. = 2  2  2 = 8

(ii) 26  11 1

(i) 199  5

(iii) 22  32  17 (iv) 109  5  22

11. Set of multiples of 6 = M(6)



{1, 2, 3, 6, 9, 18}; {1, 2, 4, 8, 16, 32};

2 30 3 15 5 5 1

3 75 5 25 5 5 1

3 135 3 45 3 15 5 5 1

HOTS WORKSHEET 1.

a) 110

b) 18

c) 1, 3, 9, 27, 81

d) 2, 3

e) 1, 3

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6th Class Mathematics

170 2.

3.

a) 42

b) 198

and (10b + a) + 18 = 10a + b

c) 120

d) 36

 9(b – a) = – 18

a)

 a – b = 2 --------- (2)

9 m ; 216 m has 24 rods, 72 m has 8 rods,

81 m has 9 rods b) 4.

Solving (1) and (2), we get a = 6 and b = 4.

Rs. 138, cheques in all 11

 The number is 46.

a)

8 11 , HCF 18 = 18 b) , HCF = 201 11 25

c)

343 , HCF = 7 197

d)

10 , HCF = 413 17

5.

a) 1

b) 2165

6.

a) 72 stones

b) 138 apples

7.

20 times

8.

60

9.

20 days

And the new obtained number is 64.

 G.C.D. of the two above numbers is ‘2’. 29. The possible sets are (2, 3) and (2, 5) Since 2 + 3 = 5 and 3 – 2 = 1 2 + 5 = 7 and 5 – 2 = 3 Therefore, these two sets will never yield a composite number. 30. Given that one of the two numbers is 25.

10. 9660 and 10080

Let the other number be x, then L.C.M = 5x. But L.C.M × H.C.F = x × 25

IIT JEE WORKSHEET

5x × H.C.F. = x × 25 1. C

2. B

3. A

4. B

5. A

6. B

7. B

8. C

9. D

10. A

11. B

12. D

13. C

14. A

15. A

16. B and C

17. A and C

18. B,C,D

19. A,B,C,D

20. A,D

21. B,C,D

22. 18

23. 188

 H.C.F. = 5. 31. A: s B: q, t C: t D: p (or) q (or) r (or) s (or) t

FRACTIONS

FORMATIVE WORKSHEET

24. 36 = 3  2  3  2 ; 72 = 3  2  3  2  2 ; 90 = 3  3  5  2 25. 18

1.

26. 3240

equivalent fractions by the same whole number.

27. N = H.C.F of [(4665 - 1305), (6905 – 4665)

6 6  3 18   11 11  3 33  a = 18

and (6905 – 1305)] = H.C.F of 3360, 2240 and 5600 = 1120

6 6  6 36   11 11  6 66  b = 66

Sum of digits in N = (1 + 1 + 2 + 0) = 4. 28. Let the unit digit and ten’s digit of a two-digit number be a and b respectively. Then a + b = 10 www.betoppers.com

--------- (1)

Multiply the numerator and denominator of each

2.

(a)

1 1 = 5 5

Number System – II Solutions

171

5 by 5, so that both fractions have 25 the same denominator.

(b) Improper fractions:

Divide

5 55 1 = = 25 25  5 5

11 7 5 101 4 9 , , , , , 2 3 5 10 3 4 6.

(a)

Both the denominators and numerators are

19  19  5 5

3 5 19 15 4

the same. 

=3

1 5 and are equivalent. 5 25

3 3 = 8 8

(b)

(b)

10 by 2, so that both fractions have 16 the same denominator.

30 = 30  6 6

7.

10 10  2 5 = = 16 10  2 8 The denominators are same, but the numerators are different.

3.

3 10 and are not equivalent. 8 16

15 5 15  5  = 25 25 25 4

20 4 = = 25 5 5 8.

1 2 5 4 3 5

The LCM of 5 and 7 is 35.

1 3

= (5 + 4) +  

3 3 7 21    5 5 7 35 4 4 5 20    7 7 5 35 Since



4.

21 20  4  , then  35 35 5 7

=9+

56 15

=9+

11 15

=9

4 3 is smaller than . 7 5

2  5

11 15 (or)

The LCM of 2, 6 and 9 is 18.

1 3

5 4

1 1 9 9    2 2 9 18

5.

5 6 30 30 0

=5

Divide



4 5

2 5

8 8 2 16    9 9 2 18

=

16 22  3 5

5 5 3 15    6 6 3 18

=

(16  5)  (22  3) 15

9 15 16   , the fractions arranged in Since 18 18 18 1 5 8 ascending order are , and . 2 6 9

=

80  66 15

=

146 11 9 15 15

(a) Proper fractions:

3 5 9 22 , , , 7 9 16 23

9.

3+

7 1 5 8 3

7 1   8 3

= (3 + 5) 

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6th Class Mathematics

172 =8+

7  3 1 8  8  3 3 8

7 6  = = 21 days 21 1

=8+

21  8 24

Therefore, the time Anandini would take to complete the whole project = 21 days.

=8+

29 24

=8+1 =9 10. 13

3

1  14  2 3   5  15  3

15.  3

5 24

 11 16  14    3 5  15

=

5 24

 55  48  14   15  15

=

2 1 +7– 3 4

1

2 1    3 4

= (13 + 7) +  = 20 +

=

83 12

= 900 The number of Indian workers =

2 2 1 litres + 37 litres – 56 litres 5 3 9 2 2 1 + 37 + – (56 + ) 5 3 9



= 2,900

 18 30 5      45 45 45 

CONCEPTIVE WORKSHEET 1.

43 litres 45

5 6

0

1

7 1  2 6

1

2

(iv)

0

1

2

3 7 3

9

1 1 7 of the project  3 days = days 6 2 2

www.betoppers.com

0

9 8

3 7 27 3 7 49 9     = = 4 18 6 2 49 7 56 4 18 27

 The whole project, 1  

1

(iii)

Number of female workers = 150 – 110 = 40 1

1 4

(ii)

10 11  150 12. Number of male workers = = 110 15 1

14.

(i)

0

43 1 litres of oil now..  It contains 11 45

13.

The total number of workers = 1,500 + 900 + 500

= 11 + 

1

5 × 900 9

= 500

2 2 1 = (30 + 37 – 56) +     5 3 9

= 11

3 × 1,500 5

The number of Chinese workers =

5 = 20 12

= 30 +

1 2

16. The number of American workers = 1,500

5 = 20 + 12

11. 30

1

7 15  = 15 1 14 2

(v)

0

1

2

3 15 7

2.

(i) 22  3

3.

(i)

1 4

(ii) 24  3 (ii)

1 4

(iii) 24  5 (iii)

9 2

Number System – II Solutions

4.

(iv)

5 11

(v)

(vii)

3 10

(viii)

(i)

7 12

173

4 5 3 5

(vi)

2 3

(ix)

9 10

1 6

(ii)

4 5

(iii)

15 96

(vi)

(iv)

25 22

(v)

(vii)

16 15

(viii) 16

3 4

5.

(x)

7. 8.

9.

(v) 23

1 1 2 3 , , , 15 6 5 4

(iii)

11 23 1 4 , , , 28 56 2 7

(v)

1 1 4 5 , , , 6 2 5 6

5 10

1 5 2 3 , , , 6 12 3 4

(ii)

1 5 4 3 , , , 3 12 7 5

(iv)

3 5 1 1 , , , 8 16 4 8

(ii)

2 1 1 1 , , , 5 3 4 9

(iii)

1 1 1 1 , , , 2 3 4 5

(iv)

4 2 5 3 , , , 5 3 8 10

(v)

47 23 7 3 , , , 50 25 10 5

(v)

23 21

(vi)

9 10

SUMM ATIVE WORKSHEET

97 70

(vii)

91 12

(ix)

1427 42

1.

3 4

2.

11 24

3.

7 11

4.

3 4

(xi)

3 56

(xii)

1 2

5.

2 7

6.

3 5

7.

1 5

8.

3 49

(xv)

99 70

29 72

(xiii)

172 15

(xiv)

13 6

(xvi)

7 24

(xvii)

725 72

(xviii)

13 12 Varied solutions

9. 5

10. 36 inches

HOTS WORKSHEET

149 20

203 (xix) 20 6.

1 3

6 11

(iv)  (vii)

6 (iii) 11

(ii) 0

12. (i)

13. (i)

(ix) 6

949 (x) 8 5 (i) 7

(iv) 8

21 20

1. A

2. A

3. D

4. D

5. A

6. D

7. C

8. D

9. A

10. C

11. D

(i)

(ii)

48 (i) 5

4 (iii) 21

33 (iv) 68

1. C

2. B

3. A

4. B

1 28

(viii) 32

5. A

6. C

7. C

8. C

9. B

10. B

IIT JEE WORKSHEET

(ii) 14

(v)

1 8

(vi)

1 5

(ix)

34 35

(x)

231 43

(vii)

(i) 33

(ii) 10

(v) 6

(vi) 3, 22, 198, 36

192 10. (i) 30 (iv)

421 20

11. (i) 12

10 25

(iii) 17

34 (ii) 9 (v)

(iv) 8 (vii) 9, 28, 108

122 (iii) 11

25 3

(ii) 3

4 11

(iii) 2

18 19

11. B,C,D

12. A and C

13. A and B

14. C and D

15. A,B,C and D

16. C and D

17. (A) 3840 km 18. (B) 1440 km 19. (A) 2400 km

20. (C)120 km

21. 1

23. 2

22. 1

24. 5

25. A– s ; B – p ; C – q ; D – t www.betoppers.com

6th Class Mathematics

174 6.

(a)

DECIMALS

87.4592 = 87 (to nearest whole number) Digit 4 < 5. Therefore, keep digit 7 and omit all digits after 7.

Place to round off.

(b)

87.4592 = 87.5 (1 d.p.)

FORMATIVE WORKSHEET

Digit 5 = 5. Therefore, add 1 to 4 and omit all digits after 4.

Place to round off.

1.

(a)

7 = 0.7 10

(b)

34 = 0.34 100

(c)

65 (c) 5 = 5.065 1, 000

2.

1 (a) = 1  8 8

Place to round off.

(d)

15 (b) = 15  8 8 1.8 7 5 8 15.0 0 0 8 70 64 60 56 40 40

0.125 8 1.000 8 20 16 40 40

1 = 0.125  8 3.

(a) 0.02 =

7.

Digit 3 < 5. Keep digit 5 and omit all digits after 5.

Place to round off.

Insert two zeros. 1

3 1 7 + 1 8 2 9 75

(b) 0.075 = 1, 000

Compare their valuesin order from left to right 485.760   In 485.760, thedigit in the tenths placeis7, 485.670  whilein 485.670, thedigit in the tenth placeis 6.

Add from right to left.

3.5  4   7.029  7  to the nearest whole number 18.953  19

The sum of these 3 decimal numbers is about 30.  The solution of 29.482 is reasonable.

9.

Total amount paid = Rs.15.95 + Rs. 22.29 + Rs. 1.65 1

1

Since, 0.7 > 0.6, therefore 485.760 is greater.

15.95

First, determine the value of each portion.

22.29

 p = 4 + 2(0.25)

Line up digits of the decimal numbers according to their place values

3.5 + 7.029 + 18.953 = 29.482

Arranged the decimals according to their place values:

.5 0 0 .0 2 9 .9 5 3 .4 8 2

Align the decimal points in a straight line.

57 57 =1 250 250

4 – 3.75 = 0.25

Digit 2 < 5. Therefore, keep digit 9 and omit all digits after 9.

8.

228 =1+ 1, 000

5.

87.4592 = 87.459 (3 d.p.)

1 3 . 5 3 9 = 13.5

(c) 1.228 = 1 + 0.228

4.

Digit 9 > 5. Therefore, add 1 to 5 and omit all digits after 5.

Place to round off.

15 = 1.875  8

2 100

=1+

87.4592 = 87.46 (2 d.p.)

+

1.65 39.89

= 4 + 0.5 p = 4.5

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 Total amount paid = Rs. 39.89

Number System – II Solutions

175 48.38 × 1 000 = 48 380

10.

(3 zeros) Move the decimal point 3 places to the right. Add 1 zero to fill up the empty space.

7 11

49.81 10.19

(c)

5 12

39.62 3.54

14. (a)

36.08

9 × 0 . 1 = 0.9 (1 d.p.) Move the decimal point 1 place to the left.

Align the decimal points.

(b)

49.81 – 10.19 – 3.54 = 36.08

7.5 × 0.01 = 0.075 (2 d.p.)

11. Amount of money left

Move the decimal point 2 places to the left.

(c)

= Rs. 2 150 – Rs. 75.89 – Rs. 186.99

2

0 14

9

1

5

0

7



9

10

.

0

0

5

.

8

9

2

0

7

4

.

1

1

 1

1 8

8 8

6 7

. .

9 1

9 2

89.4 × 0.001 = 0.0894 (3 d.p.) Move the decimal point 3 places to the left.

15. Total mass = 5.67 kg × 65.67 = 34.02 kg ×

Amount of money left = Rs. 1 887.12

6

34.02

12. (a)

18.13  (2 d.p) ×

5  (0 d.p)

90.65  (2 + 0 = 2 d.p.)  18.13 × 5 = 90.65

(b)

4.5 6  (2 d.p) ×

5.6  (1 d.p.)

16.

Align the decimal points.

15.076 5 75.380 5 25 25 38 35 30 30

Zero is added to complete the division.

 75.38  5 = 15.076

17. (a) 79.88 ¸ 100 = 0.7988

2736

(2 zeros) Move the decimal point 2 places to the left.

2 280 2 5.5 3 6  (2 + 1 = 3 d.p.)

(b) 22.128 ¸ 0.001 = 22 128 (3 decimal places) Move the decimal point 3 places to the right.

 4.56 × 5.6 = 25.536

18. (a)

13. 0.054 × 10 = 0.54

(a)

Dividend: Move the decimal point 3 places to the right, same as divisor.

(1 zero) Move the decimal point 1 place to the right.

1.796 × 100 = 179.6

(b)

5.6

= 5600 ¸ 8

0.008 = 700

(2 zeros) Move the decimal point 2 places to the right.

Divisor: Change to whole number, 8, by moving the decimal point 3 places to the right.

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6th Class Mathematics

176 700 8 5600 56

SUMM ATIVE WORKSHEET

0.03 6 7  0.18  (b) 0.18  (The reciprocal 7 61  6 7 of is ) 7 6 = 0.21

1. B

2. C

3. D

4. D

5. D

6. C

7. D

8. C

9. B

10. B

11. C

12. D

HOTS WORKSHEET

19. Wheat flour received by Aunt A = 11.775 kg  3

1. A

2. D

3. B

4. A

= 3.925 kg

5. A

6. A

7. A

8. C

9. A

10. A

11. A

12. A

Amount of wheat flour for each cake = 3.925 kg  5 = 0.785 kg

IIT JEE WORKSHEET CONCEPTIVE WORKSHEET 1.

(i) 0.07

2.

(i)

3.

4.

5.

3 (iii) 4

(ii) 0.023

3 5

(ii)

13 100

(iii)

17 40

1. B

2. B

3. C

4. A

5. A

6. D

7. A

8. A

9. A, B and C

10. B, C

11. B, C and D

12. A, B and D

(i) 1.7  1 

7 7 1 10 10

13. (B)

7 = 0.7 10

(ii) 2.3  2 

3 3 2 10 10

14. (C)

1 =1  8 8

(i) 0.2

(ii) 0.82

(iii) 1.4

(iv) 9.25 (v) 14.125

8.16

8.17

8.18

8.19

8.20

8.21

0.125 8 1.000 8 20 16 40 40

8.22

+0.01 +0.01 +0.01 +0.01 +0.01 +0.01

6.

(i) 2.143 (ii) 0.02

(iii) 52.2 (iv) 1.05 

7.

(i) 0.22, 0.39, 0.42, 0.5

15. (C) 5

65 = 5.065 1, 000

16. (D) 1

57 250

(ii) 1.01, 0.92, 0.63, 0.42

17. (A) 9

45 100

(iii) 10.1, 3.92, 0.99, 0.097

18. 5

(ii) 0.0099, 0.03, 0.042, 0.9 (iii) 5.099, 5.34, 5.43. 7.02 8.

9.

1 = 0.125 8

(i) 3.21, 3.12, 3.1, 3.09

(i) 0.42

(ii) 3.2

(iii) 9.19

(iv) 0.919

(v) 6.200

(vi) 0.150

(vii) 14.18

(viii) 4.1

(ix) 16.0

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19. 6

20. 5

21. 2

22. A – t ; B – r; C– p; D – q

Number System – II Solutions

177

SQUARES AND SQUARE ROOTS

of decimal places (d.p.) of the decimal number which is being squared.

FORMATIVE WORKSHEET 1.

5.

(a) (2.893)2  32  (2.893 rounded off to the nearest whole number is 3)

(a) The square of 9 = 9 × 9

 3×3

(b) The square of – 14 = (–14) × (–14) (c) The square of

 9

 3  3 3 =     7 7 7

(b) (698.6)2  (700)2  (698.6 rounded offf to the nearest whole number is 700)

 5  5 5 (d) The square of – =        8  8  8 (e) The square of 0.5 = 0.5 × 0.5 (f) 2.

 4,90,000

The square of –2.7 = (–2.7) × (–2.7)

(c) (0.75)2  0.82  (0.75 rounded off to the nearest whole number is 0.8)

(a) (–13) × (–13) = (–13)2  2  2  2 (b)             5  5  5 (c) 3.8 × 3.8 = (3.8)2

3.

 700 × 700

 0.8 × 0.8

2

 0.64

6.

(a) 30 < 35.8 < 40  (determine the range)

(a) (15)2 = 15 × 15 (30)2