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2 hsplkidz.com Henu Studio Pvt. Ltd. I-1654, Chittranjan Park, New Delhi (INDIA) Phone: , , henumehtani@gmail.com Website: Published in India by Eduline Publishers The moral rights of the author/s have been asserted. First Published in 2014 Reprint 2015, 2016, 2017 ISBN: Content designed and developed by Henu Studio Pvt. Ltd. Artwork and layout by Henu Studio Pvt. Ltd. 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 permission of the publisher, nor be otherwise circulated in any form of binding or cover other than that in which it is published and without a similar condition being imposed on the subsequent purchaser.

3 PREFACE Aligned to the National Curriculum Framework 2005, Eduline Look Inside Math has a vision. It focuses on instilling in children an understanding of the importance of Mathematics in their daily lives and the practical use of this knowledge. Based on this vision we present Look Inside Math, a series of three books for classes 6 to 8 for the school curriculum that is interactive and teaches core mathematical skills. The books are based on the Continuous and Comprehensive Evaluation (CCE) approach. They invite the participation of the learners, engaging them all the way, and offering a sense of success through an activitybased approach. Key features: Looking back and Warm up give lessons a head start Interactive and activity-based approach Lucid, graded, sequential presentation of mathematical concepts Exhaustive range of practice exercises and drills Clear layout and relevant illustrations to encourage understanding of abstract mathematical concepts Scope for CCE Engaging lab activities Higher Order Thinking Skills (HOTS) sections Solved questions for assistance It s time to move on with Math Asish Kumar

4 Contents Knowing Our Numbers... 7 Looking Back 7 Place Value 7 Face Value 8 Expanded Form 8 Indian and International Systems of Numeration 9 Comparing Numbers 9 Ascending and Descending Orders 10 Formation of Greatest and Smallest Numbers Using the Given digits 11 Estimation 11 Round off to the Nearest Tens/Hundreds/Thousands/Ten thousands 12 BODMAS 15 Roman Numerals 16 Playing with Numbers...21 Use of Brackets and Simplification of Brackets 21 Factors and Multiples 23 Odd and Even Numbers 23 Test of Divisibility 23 Prime and Composite Numbers 26 Co-prime Numbers 28 Twin Prime Numbers 28 Common Factors and Common Multiples 29 Prime Factorization 29 Highest Common Factor and Lowest Common Multiple (HCF and LCM) 30 Natural Numbers and Whole Numbers...41 Properties of Whole Numbers 44 Pattern in Whole Numbers 47 Negative Numbers and Integers...55 What is Negative? 55 Natural Numbers 56 Whole Numbers 56 Integers 56 Absolute Value of an Integer 57 Four Basic Operations of Integers 58

5 Fractions...65 Types of Fractions 68 Equivalent Fractions 70 Comparison of Fractions 73 Addition and Subtraction of Fractions 77 Decimals...86 Place Value Chart 88 Equivalent Decimals 89 Like and Unlike Decimals 90 Comparison of Decimals 90 Practical Application of Decimals in Daily Life 94 Addition and Subtraction of Decimals 95 Introduction to Algebra Introduction to Algebra 102 What is Algebra? 103 Like and Unlike Terms 104 Coefficient 104 Four Basic Operations in Algebra 105 Algebra as Generalization 106 Linear Equations Linear Equation 112 Solving an Equation 113 Ratio and Proportion Equivalent Ratios 121 Comparison of Ratios 122 Proportion 123 Unitary Method 127 Geometry Closed and Open Figures 134 Complex Closed Figures 135 Polygons 136 Angles 138 Triangles 141 Quadrilateral 145 Convex and Concave Quadrilaterals 147

6 Introduction of Circles 149 Concentric Circles 151 Measurement of Line Segments Relating Units of Length 158 Comparison of Line Segments 160 Pairs of Lines 165 Classification of Triangles 167 Facts About Triangles 169 Quadrilateral Parallelogram 174 Rectangle 174 Rhombus 175 Trapezium 175 Isosceles Trapezium 175 Kite 175 Polygons 176 3D Shapes 179 Nets of Some Solid Shapes 183 Symmetry Symmetrical Figures/Alphabets 188 Geometrical Construction Basic Geometrical Figures 194 Angles 199 To Bisect a Given Angle 201 Perimeter and Area Perimeter 208 Area of Plane Figures 212 Formula for Finding the Area of Rectangle and Square 213 Data Handling What is data? 219 Types of Data 220 Hypothesis 220 Collecting and Organizing Data 221 Bar Graph 228

7 1 Knowing Our Numbers Looking Back In our earlier classes we have learnt about numbers. We learnt how to name a particular number both in Indian and International systems. While writing in figures we have gathered knowledge where to put commas both in Indian and International systems of numeration. We also know how to put any number in a place value chart in both Indian and International systems. We know about face value of any number. We can arrange the numbers in descending and ascending orders. Our knowledge in comparison of numbers helps us to identify the smaller and greater numbers. We have studied about formation of greatest and smallest number using the given digits. We have learnt to solve word problems involving addition, subtraction, multiplication and division of numbers. We know about Roman numbers. We have learnt about rounding off a number to nearest tens, hundreds or thousands. Let us revise once more what we have learnt in earlier class. Place Value If a number consists of more than one digit, each digit has a value depending upon the position it occupies. Example: Find the place value of each digit of number Solution: Place the number in place value chart. Crores Lakhs thousands Ones Ten Crores Crores Ten Lakhs Lakhs Ten Thousands Thousands Hundreds Tens Ones X X Place value of digit 9 9 is placed in the Ten Lakhs column. So the place value of 9 is 9 10,00,000 = 90,00,000. Place value of digit 8 8 is placed in the Lakhs column. So the place value of 8 is 8 1,00,000 = 8,00,000. 7

8 Face Value Place value of digit 7 7 is placed in the Ten Thousands column. So the place value of 7 is 7 10,000 = 70,000. Place value of digit 6 6 is placed in the Thousands column. So the place value of 6 is 6 1,000 = 6,000. Place value of digit 5 5 is placed in the Hundreds column. So the place value of 5 is = 500. Place value of digit 4 4 is placed in the Tens column. So the place value of 4 is 4 10 = 40. Place value of digit 3 3 is placed in the Ones column. So the place value of 3 is 3 1 = 3. Face value of any digit is the value of the digit itself and does not depend upon the position it occupies. Face value of 9 is 9 Face value of 5 is 5 Face value of 8 is 8 Face value of 4 is 4 Face value of 7 is 7 Face value of 3 is 3 Face value of 6 is 6 Expanded Form The sum of place value of the digit in a number is known as the expanded form of that number. Example: Draw a place value chart of the number. 8 ten thousands thousands Hundreds tens Ones , , = 70, is the expanded form. Exercise Find the place value of each digit of the following. a) b) c) Write the expanded form of the following numbers. a) b) c) Write the short form of the following. a) b) c)

9 Indian and International systems of Numeration Indian system of numeration has places like Crores, Lakhs, thousands, Hundreds, tens and Ones. Any number can be understood easily if we write the number with commas in appropriate positions. Example: Number As per the Indian system of numeration we put commas 12,34,56,789 We read the number as Twelve crore thirty-four lakh fifty-six thousand seven hundred eighty-nine. Now we have found that it becomes easier for us to understand when we have separated the periods with commas in appropriate positions. International system of numeration has places like Millions, Thousands, Hundreds, Tens and Ones. Example: Number As per the International system of numeration we put commas 987, 654, 321 We read the number as Nine hundred eighty-seven million six hundred fifty-four thousand three hundred twenty-one. Comparing Numbers There are two rules for comparing numbers. Rule 1: If the number of digits is different in the numbers, the number having more digits is greater. Example: Compare numbers 2597, 28437, Here 2597 has four digits have five digits have six digits. So, > > Rule 2: If the number of digits is same in the numbers, then the following steps must be observed. Example: Compare the numbers 65789, 65795, All the numbers have five digits Start comparing from left All numbers have 6 in the first place same. All numbers have 5 in the second place same. All numbers have 7 in the third place same. Number Digit 8 in the fourth place Number Digit 9 in the fourth place Number Digit 8 in the fourth place So we find that is greater than both and

10 10 Therefore is the greatest. Number Number Has digit 9 in the fifth placed Has digit 4 in the fifth placed So, number > Therefore, > > Ascending and Descending Orders Ascending order Arranging numbers from smallest to greatest. Descending order Arranging numbers from greatest to smallest. Example: Arrange the following numbers both in ascending and descending orders , 3729, , digits digits digits digits Number with four digits is the smallest. Number having five digits ending with 0 is smaller than number having five digits ending with 9 since all other digits are same. Number having six digits is the greatest. Therefore in ascending order smallest to greatest 3729, 49750, 49759, descending order greatest to smallest , 49759, 49750, 3729 Exercise Draw a place value chart of the following numbers in Indian system of numeration. a) 8472 b) c) Draw a place value chart of the following numbers in International system of numeration. a) 3487 b) c) Put commas in appropriate positions for the following numbers in Indian system and write their number names. a) b) c) Put commas in appropriate positions for the following numbers in International system and write their number names. a) b) c) Find the place value and face value of the encircled digit. a) b) c)

11 Formation of Greatest and smallest Numbers Using the Given digits To form the greatest number using the given digits, we arrange the digits in descending order. To form the smallest number using the given digits, we arrange the digits in ascending order. But if one of the given digits is 0, then while forming the smallest number, write 0 in the second place from the left. Example 1: Write the greatest and smallest number using the digits 9, 6, 8, 5, 2, 4. Greatest number Write the digits in descending order Smallest number Write the digits in ascending order Example 2: Write the greatest and smallest number using the digits 6, 7, 0, 8, 3, 9. Greatest number Write the digits in descending order Smallest number Write the digits in ascending order but keep 0 in 2 nd place from the left Estimation In our daily life, we make an estimate in money, numbers, times, etc. For example, as per any purchases we carry money accordingly. Actual calculation will take long time and approximation will save time. The figures may not be actual but near about. This process of calculation is called estimation. Example: Supposing we wish to purchase 1 packet tea and biscuit, 1 litre milk, 1 dozen egg and 1 packet bread. How much money we should carry? Quickly, we calculate by approximation. Estimated cost Actual cost Tea ` ` Biscuit ` ` Milk ` ` Egg ` ` Bread ` ` ` ` The actual amount of ` is very near to our approximation of ` Exercise Estimate the following numbers to the nearest tens and thousands. a) b) c) d) Estimate the following difference to the nearest thousands. a) b) c) d) Estimate the difference of the following numbers to the nearest thousands. Find the actual difference. a) and b) and

12 Round off to the Nearest Tens/Hundreds/Thousands/Ten thousands Round off to the nearest tens Let us take the numbers 12,24,35,57. Number 12 Lies between 10 and 20 Number 12 is near to number 10. So, 12 is rounded off to Near 12 Far Number 24 Lies between 20 and 30 Number 24 is near to number 20. So, 24 is rounded off to Near 24 Far Same Same Number 35 Lies between 30 and Just in between 30 and 40. It is accepted that 5 is rounded off to next tens, i.e., 40. So 35 is rounded off to 40. Far Near Number 57 Lies between 50 and Number 57 is near to number 60. So it is rounded off to 60. Rounded off to the nearest hundred Look at the digit in tens place. If it is 0, 1, 2, 3 or 4 simply put zeros in the tens and ones places. If the digit at tens place is 5, 6, 7, 8 and 9, put zeros in one place and tens place, and increase the digit at hundreds place by 1. Examples: 1. Round off 2724 to the nearest hundred Digit at tens place is 2 so put zeros in the tens and the ones places. Rounded off to the nearest hundred would be Round off 5857 to the nearest hundred Digit at tens place is 5 so put zeros in ones and tens places and increase the digit at hundreds places by 1. Rounded off is Round off 7673 to the nearest hundred Digit at tens place is 7 so put zeros in ones and tens places and increase the digit at hundreds place by 1. Rounded off is

13 Round off to the nearest thousand Look at the digit at hundreds place. If it lies between 0 to 4, we put zeros at ones, tens and hundreds places. If the digit at hundreds place lies between 5 to 9, we put zeros at ones, tens, hundreds places and increase the digit at thousands place by 1. Examples: to be rounded off to the nearest thousand Digit at hundreds place is 4 Put zeros at ones, tens and hundreds places. Rounded off number would be to be rounded off to the nearest thousand Digit at hundreds place is 8 Put zeros at ones, tens and hundreds places and increase the digit at thousands place by 1. Rounded off number would be Rounded off to the nearest ten thousands Look at the digit at thousands place. If it lies between 0 to 4 Put zeros at ones, tens, hundreds and thousands places. If it lies between 5 to 9 Put zeros at ones, tens, hundreds and thousands places and increase the digit at ten thousands place by 1. Examples: to be rounded off to the nearest ten thousands The digit at thousands place is 4 Put zeros at ones, tens, hundreds and thousands places Rounded off number would be to be rounded off to the nearest ten thousands The digit at thousands place is 8 Put zeros at ones, tens, hundreds and thousands places. Increase the digit at ten thousands place by Rounded off number would be Exercise Compare the following numbers using symbols (>, <, = ). a) and b) and c) and Form the greatest and smallest numbers using the following digits. a) 4, 3, 0, 1, 2, 3 b) 2, 0, 1, 7, 5, 8 c) 6, 0, 2, 3, 4, 1 3. Round off the following numbers to the nearest ten. a) 794 b) 7861 c)

14 4. Round off the following numbers to the nearest hundred. a) 569 b) 7865 c) Round off following numbers to the nearest thousand. a) 5721 b) c) Round off the following numbers to the nearest ten thousand. a) b) c) Word Problems 1. During a hockey match between India and Holland initially 25,367 people were present to watch the game. As the match progressed, since India was losing 15,498 people left the ground. How many people watched the game till the end? Solution: Number of People present at the start of the game Number of People left in between Number of Subtraction has to be done Answer 9869 Answer: People present till end of the match is Mani purchased 54 chocolates costing ` 125 each, 72 patties costing ` 45 each, and 24 packets of potato chips ` 78 each. How much money Mani needs to buy all the items? Solution: Money spent on Chocolates Money spent on Patties ` Money spent on Potato chips (+) ` (+) ` Total money required ` 11,862 Answer: Mani needs ` 11,862 to purchase the items.

15 3. A box of ice cream has 4 rows, in each row there are 12 ice creams. The total ice creams are to be divided equally among 8 children. How many will each child get? Solution: 4 rows, in each row there are 12 ice creams. Total number of ice creams 4 12 = ice creams are to be divided among 8 children. Each child will get 48 8 = 48/8 = 6 ice creams. Each child will get 6 ice creams. Exercise Raju deposits ` 4, 24,759 in January in his bank a/c, ` 27,924 in February and withdraws ` 75,215 in March. What amounts is left in the bank at the end of March? 2. Rohit purchases 49 pencils each costing ` 12, 27 rubbers each costing `8 and 15 sharpeners each costing ` 10. What is the total amount required by Raju to buy the above items? 3. Neel goes to market and purchases 15 packets of biscuits. Each packet contains 10 biscuits. Neel wants to distribute the total biscuits to his 25 friends. How many biscuits will each friend get? BODMAs While applying all the operations brackets, of, addition, subtraction, multiplication and division together, we follow BODMAS rule. B Brackets O Of D Division M Multiplication A Addition S Subtraction We have to remember the following order of operations: 1st operation Brackets, remove brackets 2 nd operation Of, Multiplication of two numbers having of between them 3 rd operation Division 4 th operation Multiplication 5 th operation Addition 6 th operation Subtraction Example: Simplify the following: (27 9) - 12 of 3 1 st step (B) Remove bracket of 3 2 nd step (O) Do of operation rd step (D) Division th step (M) Multiplication th step (A) Addition th step (S) Subtraction 1 15

16 Exercise 1.6 Simplify the following: of 13 + (15-8) ( ) of 15 + (19-8) (10 + 2) ( 3 15) + (19-12) 7 10 Roman Numerals In England, kings used this system of naming their kings that carried the same surname. The Romans in their time conquered many different countries and England was one such country. After conquering England, the Romans settled there, where the English and other people would adapt the number system used by the Romans. This is why for example kings of England like King Henry the 1st written as King Henry I (pronounced: as King Henry the first) or king Henry the 8th written as Henry VIII (pronounced: King Henry the eighth) used these Roman numerals in their names. Traditionally Roman numerals were used to indicate the order of family offsprings of the same name (For example, II was used instead of Jr., III for the third, and IV for the fourth and so on) and also if the same surname were used by church leaders (Popes) and kings (whether or not by offspring) this numbering system was used to indicate that order. Roman numerals didn t have any symbol for zero (0) and as a result, because the concept of zero did not exist, the numeral placement was sometimes based on subtraction rather than addition. The largest number that could be represented by the Roman numerals system using their rules was 4,999. There are seven Roman symbols (I, V, X, L, C, D, M). These seven symbols along with their corresponding values in the Hindu-Arabic system are given below. 16 I V X L C D M Roman numerals are formed by using these seven symbols in different coordinations. Of course, certain rules are to be followed: Rule 1: Only I, X, C and M can be repeated but not more than 3 times. V, L and D cannot be repeated. Repetition of certain symbols means addition. Example: III = 3 XXX = 30 CCC = 300 MMM = 3000

17 Rule 2: A smaller symbol written to the right of larger symbol is always added to the larger symbol. Example: VIII = 8 XV = 15 CLX = 160 Rule 3: If a symbol of smaller value is written to the left of larger value symbol, then the value of smaller symbol is subtracted from the value of larger symbol. Example: IX 10-1 = 9 XC = 90 CM = 900 Rule 4: If a smaller value symbol is placed in between two greater value symbols, then it will be subtracted from the larger value symbol immediately following it. Example: XIV 10 + (5-1) = 14 CXC (100-10) = 190 XXIV (5-1) = 24. Exercise Write the following Roman numerals in Hindu-Arabic numerals. a) LXIV b) CDLXV c) CMLIX d) DCCXLVI e) CCCLIX 2. Write the following Hindu-Arabic numerals in Roman numerals. a) 985 b) 489 c) 786 d) 627 e) 577 Fill in the blanks. 1. IV + X = 2. L = X 3. C IX + VII = 4. CDXXI = 5. MCMXC = 6. CCVII = 7. MLXVI = 8. MMXIII = 9. MMVIII = 10. MCMLIV = WORKsHEEt 17

18 REVIsION EXERCIsE 1. Find the place value and face value of the encircled digit. a) b) c) Write the expanded form of the following numbers. a) b) c) Draw the place value chart for the following numbers in both the Indian and International systems of numeration. a) b) c) Put commas in appropriate positions for the following numbers both in the Indian and International systems and write their number names. a) b) c) Compare the following numbers (use symbols >, < =). a) , b) , c) , Form the greatest and smallest numbers using the following digits. a) 4, 5, 6, 7 b) 8, 0, 5, 9 c) 2, 0, 5, 8 7. Round off the following numbers to the nearest ten. a) 349 b) 6561 c) Round off the following numbers to the nearest hundred. a) 458 b) 9865 c) Round off following numbers to the nearest thousand. a) 1292 b) c) Round off the following numbers to the nearest ten thousands. a) b) c) Estimate the following numbers to the nearest ten thousands. a) b) Estimate the following difference to the nearest hundreds. a) b) Ramesh deposits ` 1, 34,523 in January in his bank account, ` 87,521 in February and withdraws ` 35,598 in March. What will be left in the bank at the end of March? 14. Reema purchases 34 protractors each costing ` 16, 27 pens each costing ` 23 and 15 rubbers each costing ` 8. What is the money required by Reema to buy the above items? 15. Neela goes to market and purchases 25 packets of biscuits. Each packet contains 20 biscuits. Neela wants to distribute the total number of biscuits to her 20 friends. How many biscuits each friend will get? 16. Simplifying the following: a) (3 5) b) 120 (45 9-6)

19 Rule: WORKsHEEt 1. Make 5 numbers by interchanging the place value of the number given. 2. Express the numbers in words in Indian system. 3. Express the numbers in words in International system. Numbers indian Number System international number system ,16,789 Twenty five lakh sixteen thousand seven hundred eighty nine 2,516,789 Two million five hundred sixteen thousand seven hundred eighty nine. REFLECtiON Clue 1: The number is greater than [(5 5) - 5] Clue 2: The number is less than [100 - (5 5)] Clue 3: The number is an even number Clue 4: If you count by 5 s, you do not say the number Clue 5: The sum of the digits is 8 Clue 6: Both digits are same. Sahiba had a dream in which she was selling diamonds on an international market. She started out with 120 diamonds. An Egyptian bought Ç diamonds i.e. 15 diamonds A Babylonian bought diamonds i.e. 36 diamonds A Roman bought XXIV diamonds i.e. 24 diamondsds A Hindu-Arabic person bought the remaining diamonds How many diamonds did the Hindu-Arabic person buy? 19

20 Math Lab Activity Objective: Create palindrome number Materials Required: Flash cards (0 to 9) Lesson Development: 1. Divide the class into small groups. 2. Distribute flash cards (0 9) to each group. 3. Ask students to place the cards upside down on the table. 4. Ask the students to choose randomly 5 cards. 5. Tell them to make a 5-digit number. 6. Tell them to make the greatest 5-digit number. 7. Tell them to make the least 5-digit number. 8. Ask them to add the two numbers (greatest and least). 9. Explore the number. Can you read the number same from both the direction? 10. This type of numbers is called Palindrome numbers. 11. If you don t get palindrome number then repeat the exercise. Example: original number greatest number least number 5 Addition ( ) = palindrome number

21 2 Playing with Numbers Let me tell you a story. I went to my aunt s place in summer vacation. I wanted to buy some gifts for my friends. I was short of money. My aunt told me to take the cash from her. After coming back from her place she sent me a card where she said. Pink Elephants Dislike Mice And Snails. She wrote one expression (9 8) - 5. I showed all my friends the expression and asked for the solution. My friends gave different solutions. I was totally puzzled as to which solution I should accept. I read the statement again Pink Elephants Dislike Mice And Snails. In Pink, P stands for Parenthesis, i.e. Brackets Elephants, E stands for exponent, i.e Dislike, D stands for division Mice, M stands for multiplication And, A stands for addition, i.e Snails, S stands for subtraction, i.e. 103 I have to pay ` 103 to my aunt. To remember the order of operations by using a memory devise Pink Elephants Dislike Mice And Snails Use of Brackets and simplification of Brackets Brackets are used for grouping the numbers. This is done in order to simplify any expression easily. Every number in a sum has its own positive and negative signs. We have learnt earlier that in mathematics there are four operations addition, subtraction, multiplication and division. Now, which operation has to be performed first to simplify any expression. In order to avoid confusion an International method has been accepted regarding the order of operation. 1 st operation Remove bracket Bracket is written in short form as B 2 nd operation Of Means multiplication of two numbers having of between them. In short it is written as O 3 rd operation Division In short it is written as D 4 th operation Multiplication In short it is written as M 5 th operation Addition In short it is written as A 6 th operation Subtraction In short it is written as S 21

22 So we follow rule BODMAS to simplify any expression and operations are done accordingly. there are different kinds of brackets: a) [ ] Known as square brackets / rectangular brackets / box brackets / big brackets. b) { } Known as curly brackets / braces. c) ( ) Known as simple brackets / parenthesis. Now certain orders are to be followed to remove brackets. 1 st to be solved is ( ). 2 nd to be solved is { }. 3 rd to be solved is [ ]. Example: Simplify 256 [69 {4 6 + (7 of )}] Solution: 1 st step Remove simple brackets means to perform all operation inside simple brackets. (7 of ) By applying BODMAS rules, we get ( ) = = nd step Remove curly brackets. Do all operations inside curly brackets. { } = = rd step Remove rectangular brackets. B RACKETS ( ) [ ] { } = [69-53] = 16. O F D IVIDE / Final step = 16. M ULTIPLY A DDITION + Answer: 16. S UBTRACTION - Simplify the following. 22 Exercise ( ) 2. { ( )} [69 {7 6 (12 3 8)}] 4. ( of 29) + [ 19 {385 (30 6 7)}] [ 18 of 1 ] { ( )}]

23 Factors and Multiples Factors: If two or more numbers are multiplied together each of the number is a factor of the product. Example: = 105 Factors 1, 3, 5, 7, 15, 21, 35, 105 of 105. Properties of factors: 1 is a factor of every number. Every factor of a number divides the number exactly. The number of factors of a number is finite. Multiples: The product of any two numbers is a multiple of each of the two numbers. Examples: = 56 Therefore 56 is a multiple of 7 and 8 both = = = 10, 2 10 = = 30, 4 10 = and so on So the number 10 has many multiples and it can go to infinity. Properties of multiples: Every number is a multiple of the number itself. Every number is a multiple of 1. Every number has infinite multiples. Therefore we can sum up: 1. Every number is a multiple of itself. 2. Every number is a multiple of The number of multiples of any number is infinite is a factor of any number. 5. Every factor of a number divides the number exactly. 6. The number of factors of a number is finite. Odd and Even Numbers Odd numbers: The numbers not divisible by 2 or not multiples of 2 are called odd numbers. For example, 1, 3, 5, Even numbers: The numbers which are multiples of 2 or divisible by 2 are called even numbers. For example, 2, 4, 6, test of Divisibility Divisibility test by 2 A number is divisible by 2, if the one s digit is 0, 2, 4, 6 and 8. Examples: 10 is divisible by 2, since 0 is present in one s digit. 16 is divisible by 2, since 6 is an even number present in one s digit. 8 is divisible by 2, since 8 is an even number. 23

24 Divisibility test by 3 A number is divisible by 3, if the sum of digits of the number is divisible by 3. Examples: Number 15 is divisible by 3, since sum of the digits of the number is = 6, which is divisible by 3, so the number 15 is divisible by 3. Number 984 is divisible by 3, since sum of the digits of the number is = 21, is divisible by 3, so the number 984 is divisible by 3. Divisibility test by 4 A number is divisible by 4, if the number formed by the last two digits is divisible by 4. Examples: Number 116 is divisible by 4, since the number formed by the last two digits is 16 and 16 is divisible by 4. Number is divisible by 4, since the number formed by last two digits is 48 and 48 is divisible by 4. Divisibility test by 5 A number is divisible by 5, if the number in the one s digit is either 0 or 5. Examples: Number 150 is divisible by 5 since the number in the one s digit is 0. Number 855 is divisible by 5 since the number in the one s digits is 5. Divisibility test by 6 A number is divisible by 6, if the number is divisible by 2 and 3 both. Examples: Number 234 is divisible by 6 since 234 is divisible by 2, as an even number 4 is in one s place. 234 is also divisible by 3 since the sum of the digits = 9 is divisible by 3. So 234 is divisible by 6 since it is divisible by both 2 and 3. Divisibility test by 8 A number is divisible by 8, if the number formed by last three digits is divisible by 8. Examples: Number is divisible by 8 since the number formed by last three digits, i.e. 984 is divisible by 8. Number is divisible by 8 since the number formed by last three digits, i.e. 872 is divisible by 8. 24

25 Divisibility test by 9 A number is divisible by 9, if the sum of the digits of the number is divisible by 9. Examples: is divisible by 9 since the sum of the digits of the number ( ) = 18, is divisible by 9. Number is divisible by 9 since the sum of the digits of the number ( ) = 36, is divisible by 9. Divisibility test by 10 A number is divisible by 10, if the digit in the unit s place is 0. Examples: Numbers 10, 20, 30, 40, 50, 100, 1000, all have 0 in the unit s place. So all of them are divisible by 10. Divisibility test by 11 A number is divisible by 11, if the difference between the sum of its digits in odd and in even places is either 0 or a multiple of 11. Examples: Number 28457, here sum of the digits in odd places = 13. sum of the digits in even places = 13. Difference between the sums = 0. So the number is divisible by 11. Number 30789, here sum of the digits at odd places = 19. sum of the digits at even places = 8. Difference between the sums 19 8 = 11. So the number is divisible by 11. Exercise Write all factors of: a) 36 b) 81 c) 275 d) 380 e) Write the first five multiples of: a) 25 b) 12 c) 18 d) 30 e) Write all the multiples of 13 lying between 65 and Write all the even multiples of 15 lying between 65 and

26 5. Fill in the blanks: a) A number is a multiple of. b) is a factor of every number. c) Every number has factor. d) Every number is a of itself. 6. Answer the following by Y for yes and N for no. a) Is 1 a factor of 3456? b) Is 7896 a factor of 7896? c) Is 24 a factor of 126? d) Is 18 a factor of 57? e) Is 9 a factor of 1116? f) Is 87 a multiple of 9? g) Is 21 a multiple of 7? h) Is 65 a multiple of 5? i) Is 300 a multiple of 10? 7. Find out which of the following numbers are odd: a) 37 b) 82 c) 75 d) 96 e) Apply the divisibility rules and find out the following: Say Y for yes and N for no. S. No. Number Divisible by 2 Divisible by 3 Divisible by 4 Divisible by 5 Divisible by 6 Divisible by 8 Divisible by 9 Divisible by 10 Divisible by Prime and Composite Numbers Prime numbers The numbers which have only two factors 1 and the number itself are called prime numbers. Example: 2, 3, 5, 7, 11, 13, 17, is the smallest prime number. 26

27 Composite numbers The numbers which have more than two factors are called composite numbers. Example: 4, 6, 8, 9, 10, 12 4 is the smallest composite number. A Greek mathematician Eratosthenes devised a method to find prime numbers from 1 to 100 in the third century B.C. the method is explained below: First prepare a table from 1 to Now follow the steps: Step 1. Cross 1 as it is neither a prime number nor a composite number. Step 2. Encircle 2 and cross all multiples of 2. Step 3. Encircle 3 and cross all multiples of 3. Step 4. Encircle 5 and cross all multiples of 5. Step 5. Encircle 7 and cross all multiples of 7. Now encircle the numbers which are not crossed. So, the total encircled numbers between 1 to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97 These are the prime numbers from 1 to

28 Co-prime Numbers Two numbers that have only 1 as a common factor are called co-prime numbers. Therefore pairs of prime numbers are co-prime numbers. Example: (2, 3), (3, 5), (5, 7) and so on. twin Prime Numbers Two prime numbers with a gap of only one number are called twin prime numbers. Example: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31) How to find out a number is prime or not? Let us find out whether the numbers 31 and 83 are prime or not. Step 1. Find out a square number immediately after 31. It is 36 since 36 = 6 6. Step 2. The prime numbers below 6 are 2, 3, 5. Step 3. Find out if the number 31 is divisible by 2, 3 and 5. It is not. So, 31 is a prime number. Similarly let us work out for 83. Step 1. The square number immediately after 83 is = Step 2. The prime numbers below 10 are 2, 3, 5, 7. Step 3. Find out the number 83 is divisible by 2, 3, 5 and 7. It is not. So, the number 83 is a prime number. Exercise Make a list of all the prime and composite numbers from 1 to Which number is neither prime nor composite? 3. What is the smallest prime number? 4. How many even numbers is prime? 5. What is the smallest composite number? 6. Find out the following numbers are prime or not: a) 187 b) 149 c) 277 d) Find out co-prime numbers from the following: a) (13, 23) b) (57, 84) c) (67, 97) d) (47, 68) e) (29, 83) 8. Find out the twin prime numbers: a) (5, 7) b) (37, 47) c) (71, 73) d) (79, 83) e) (43, 53) f) (41, 43) g) (11, 13). 28

29 Points to Remember 1. Every number is a multiple of The smallest multiple of a number is the number itself. 3. Number of multiples of any number is infinite is the smallest prime number is the only even prime number is neither a prime nor a composite number. 7. Smallest composite number is 4. Common Factors and Common Multiples Common factors Take two numbers 24 and 36. Find the factors of the numbers. 24 = Factors are 1, 2, 3, 4, 6, 8, 12, = Factors are 1, 2, 3, 4, 6, 9, 12, 18, 36. So the common factors of 24 and 36 are 2, 3, 4, 6 and 12. Common multiples Find the first six multiples of 4 and 6. Multiples of 4 4, 8, 12, 16, 20, 24, 28, 32 Multiples of 6 6, 12, 18, 24, 30, 36, 42, 48 First six multiples of 4 4, 8, 12, 16, 20, 24 First four multiples of 6 6, 12, 18, 24, So, the common multiples of 4 and 6 are 12 and 24. Prime Factorization Factor tree method Prime factorization is the process expressing a number as product of prime numbers only. Prime factorization of 36. So, 36 = This method is called factor tree method. 29

30 Division method We can do the factorization in another method So, 36 = This method of factorization is called division method. Point to Remember In prime factorization, all the factors are prime numbers. Highest Common Factor and Lowest Common Multiple (HCF and LCM) We know about common factors and common multiples. Now we will study about highest common factor (HCF) and lowest common multiple (LCM). Highest Common Factor Let us take two numbers 36 and 64. Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36 Factors of 64 are 1, 2, 4, 8, 16, 32, and 64 Common factors are 1, 2, and 4. Highest common factor is 4. The greatest common factor of two or more numbers is called their highest common factor (HCF). Lowest Common Multiple Let us take the same numbers 36 and 64. Multiples of 36 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, 396, 432, 468, 504, 540, 576, Multiples of 64 64, 128, 192, 256, 320, 384, 448, 512, 576, Lowest Common Multiple is 576. LCM by factorization method The prime factorization of 36 and 64 are 36 = = The common factors are

31 Factors which are not common 2, 2, 2, 2, 3, 3 We observe that the product of common factors and factors which are not common Therefore we can say that the lowest common multiple is the product of the common factors and the factors which are not common. HCF and LCM by prime factorization method Example 1: Find HCF of 36, 72 Solution: = = HCF of 36 and 72 = = 12 Example 2: Find LCM of 48, 64 and 50. Solution: = = = LCM of 48, 64 and 50 = = HCF and LCM by Division Method We can find the HCF of two or more numbers by dividing the bigger number by another number and then dividing another number by the remainder. Repeat the process of dividing the preceding divisor by the remainder till we get remainder as zero. The last divisor is the HCF of the given numbers. This method is called Long Division Method. 31

32 Example: Find the HCF of 60 and 72 by long division method. Solution: The bigger number is 72. Smaller number is 60. So, divide 72 by 60. Keep on dividing the preceding divisor by the remainder till we get remainder as zero. 60 )72 ( )60 ( The last divisor, i.e. 12 is the HCF of the numbers. So, HCF of 60 and 72 is 12. To find the LCM of two or more numbers by division method, we follow as below. 1. Write all the numbers in a row. 2. Find the least prime number which divides at least two numbers. 3. Continue the process till all the numbers in a row are no more divisible by any number. 4. The product of the prime divisors and the quotient will be the LCM. Example: Find the LCM of 48, 64 and 50 by common division dethod. Solution: 2 48, 64, 50 So, LCM = Product of divisor and quotient 2 24, 32, 25 = , 16, 25 = = , 8, 25 = , 4, 25 = , 4, 5 Relation between HCF and LCM The product of two numbers is always equal to the product of their HCF and LCM. Let us consider the numbers 24 and 48. HCF of 24 and )48 ( HCF of 24 and 48 = 24 LCM of 24 and , 48 LCM = = , 24 So, HCF LCM = , 12 Product of numbers = So, product of two numbers = HCF of two numbers LCM of two numbers. 3 3, 6 1, 2 32

33 Example 1: three containers contain 20 litres, 30 litres and 40 litres of milk. Find the maximum capacity of container which can measure the milk of three containers exact number of times. Solution: To find the exact measure of container we have to find out the HCF of 20, 30 and 40. We use division method of HCF. 20 )30 ( )20 ( )40 (4 40 HCF of 20, 30 and 40 = 10 Hence 10 litres is the maximum capacity of a container that can measure 20, 30 and 40 litres milk exactly. Example 2: the dimensions of a room is 50 cm, 75 cm and 130 cm. Find the largest measuring tape which can measure three dimension exactly. Solution: We will have to find the HCF of 50, 75 and 130. Use long Division Method 50 )75 ( )50 ( )130( )25 (5 25 HCF of 50, 75 and 130 is 5. So, the 5 cm tape can measure 50 cm, 75 cm and 130 cm exactly 33

34 Example 3: Four bells ring at an interval of 5 minutes, 10 minutes, 15 minutes and 20 minutes. All the four bells ring at 8 O clock in the morning. When will they ring again? Solution: We need to calculate the LCM of 5, 10, 15 and 20. Use Division Method 5 5, 10, 15, , 2, 3, 4 1, 1, 3, 2 LCM = = 60. The bells will ring again at an interval of 60 minutes. Therefore all the bells will ring at 9 O clock in the morning. Example 4: Determine the greatest three digit number exactly divisible by 10, 15 and 20. Solution: To find out the greatest three digit number exactly divisible by 10, 15 and 20, we will have to determine the LCM of 10, 15 and 20. Use Division Method 2 10, 15, , 15, 10 1, 3, 2 LCM of 10, 15 and 20 = = 60. Greatest 3-digit number is )999( So, = 960 is the greatest 3-digit number exactly divisible by 10, 15 and 20. Example 5: Determine the smallest 3-digit number exactly divisible by 10, 15 and 20. Solution: Find the LCM of 10, 15 and , 15, , 15, 10 1, 3, 2 LCM = = 60 34

35 Smallest 3-digit number is 100. Now we have to find out a number that should be added to 100. So that it is exactly divisible by )100( So the number divisible by 60 but greater than 100 would be (60-40) = = 120. Threfore, 120 is the smallest 3-digit number which is exactly divisible by 10, 15 and 20. Exercise Find the common factors of the following numbers. a) 35 and 55 b) 33 and 77 c) 13 and 52 d) 45 and Find first three common multiples of the following numbers. a) 6 and 8 b) 10 and 15 c) 4 and 6 d) 12 and Find out by what greatest number the following numbers are divisible. a) 12, 22 b) 20, 35 c) 45, 81 d) 64, 56 e) 81, If a number is divisible by 5 and 6, by what other number that number will be divisible? 5. Find the prime factorization of the following number. a) 360 b) 64 c) 612 d) Find the HCF of the following numbers. a) 24 and 96 b) 25 and 125 c) 30 and 90 d) 13 and Find the HCF by prime factorization and division method. a) 12, 20 and 36 b) 14, 70 and 126 c) 15, 25 and 145 d) 84, 108 and Find the LCM of the following numbers. a) 15 and 85 b) 9 and 27 c) 25 and 6 d) 13 and Find the LCM using prime factorization. a) 82, 92 b) 15, 30, 45 c) 125, 175, LCM of three numbers is 30. What are the numbers? 11. In a long jump, three boys jump together from a particular spot. One boy covers a distance of 10 cm, second covers 50 cm and third covers 70 cm. Find out the minimum distance each should cover so that all can cover the distance in complete steps. 12. The length, breath and height of a room are 60 cm, 40 cm and 50 cm. What should be the greatest length of tape which can measure the room exactly? 13. Find the greatest number which will divide 45, 72 and 999 leaving remainder as 5, 2 and 9. 35

36 14. A boy completes a 100 m circular track race in 5 min. Another boy runs in the same directions completes the race in 6 minutes. If they start from the same point then after how many minutes will they meet at the starting point. 15. Find the smallest 3-digit number which is exactly divisible by 5, 10 and 20. Points to Remember 1. A number that divides a particular number exactly is called a factor of the number is a factor of every number. 3. Every number is a factor of itself. 4. A multiple of a number is exactly divisible by the number. 5. Every number is a multiple of itself. 6. A number can have infinite number of multiples. 7. A number that has only two factors - 1 and the number itself is called prime number is the only even and smallest prime number is a unique number, it is neither prime nor a composite number. 10. A number is divisible: By 2 if the digits in one s place is either 0 or even number. By 3 if the sum of the digits of the number is divisible by 3. By 4 if the last two digits of the number is divisible by 4. By 5 if the digits in ones place is either 0 or 5. By 6 if the number is divisible both by 2 and 3. By 8 if the last three digits of a number is divisible by 8. By 9 if the sum of the digits of number is divisible by 9. By 10 if the digit in one s place is 0. By 11 if the difference between the sum of its digits in odd places and in even places is either 0 or a multiple of HCF is the largest number that divides all the given numbers exactly. 12. LCM is the smallest number which is divisible by all the given numbers. 13. The product of LCM and HCF of any two numbers is equal to the product of given numbers. 36

37 REVIsION EXERCIsE 1. Simplify the following. a) 96 [18 {63 7 (18 5 of 3)}] b) 81 of [60 {8 7 (13 5 of 2)}] c) 49 - [23 + {27 (9 of 3 27)}] d) 72 + [65 { (7 of 3-21)}] 2. Write all the factors of the following. a) 37 b) 56 c) 225 d) Write 5 multiples of the following. a) 26 b) 11 c) 13 d) 9 4. List the even prime numbers between 1 and List the odd prime numbers between 30 and Write seven consecutive composite numbers between two prime numbers from 1 to Test the divisibility of the number 7830 by 2, 3, 4, 5, 6, 8, 9 and 10. Explain reasons. 8. Complete the factor tree using prime numbers. a) 64 b) Find the HCF of the following numbers using prime factorization method and long division method. a) 20, 80 b) 13, 169 c) 5, 50, 100 d) 16, 72, Find the LCM of the following numbers by prime factorization method and long division method. a) 18, 66 b) 24, 117 c) 36, 72, 96, 108 d) 140, 180, LCM of two numbers is 120 and the HCF is 2. If one of the numbers is 10. Find the other number. 12. Two tankers contain 96 litres and 120 litres of water respectively. Find the capacity of the container that can measure the water exactly. 13. Four bells ring at the intervals of 10, 15, 20 and 25 minutes. At what time they will ring together if they ring at 7 AM? 37

38 WORKsHEEt Objective: Understand the rules if there are more than two operations in one expression. There are 50 passengers in a bus going to the mall. At one stop, 3 groups of 4 passengers leave the bus to go for shopping. Next stop, 5 groups of 4 passengers leave the bus to eat lunch. How many passengers are left in the bus? Express the problem as expression: To solve the problem correctly, follow the order of operations. Step 1. Simplify the terms within parenthesis. Step 2. Simplify terms with exponents. Step 3. Multiply and divide from the left to right. Step 4. Add and subtract from left to right. 1. Think of a number between 1 to Look at each card and say yes if your number is in the cards. 3. Take all the cards having yes response. 4. Add the lowest number in each card. 5. The number will be your number which you have thought. Example: I thought of number 63. total: = 63 Card A yes 1 Card B yes 2 Card C yes 4 Card D yes 8 Card E yes 16 Card F yes 32 Card A Card B

39 Card C Card D Card E Card F Reflection Follow the directions to simplify the expression Work inside parenthesis : Work for exponents: Work for multiplication: (25 5) 3 (18 5) (9 4) Work for addition and subtraction from left to right: 39

40 Math Lab Activity Objective: Playing with numbers Rita: Savita I can guess your birthday Savita: How, tell me! Rita: Give the numbers to months like January = 1, February = 2, March = 3, April = 4, May = 5, June = 6, July = 7, August = 8, September = 9, October = 10, November = 11, December = Multiply the number of the month in which you were born by Add Double the answer. 4. Subtract Multiply by 5 6. Subtract Double the answer. 8. Add 9 9. Add the number of the day on which you were born. 10. Subtract 203. Example: Let me try to explain how it works with a specific example. I am going to use the month as 8 and the day as 12 and go through the steps = = = = = = = = = = will be read as 12 th August. 40