Approved by the Government of Nepal, Ministry 6of Education, Science and Technology, Curriculum Development Centre, Sanothimi, Bhaktapur as an additional material.Please scan forE-book6
Publisher : Sundar Pathshala Prakashan Pvt. Ltd.Anamnagar, Kathmandu, NepalLayout Design: Sundar Pathshala Prakashan DesktopEdition : 2063 (First)2067 (Second)2080 (Third)2083 (Fourth) Revised & UpdatedPrinted in Nepal
I feel proud to present this edition of Mathematics book. It is based on the latest syllabus, formed by Curriculum Development Centre (CDC). I have emphasized the theoretical as well as the numerical aspects of the mathematics course. The underlying concepts have been gradually and systematically developed.In each chapter, all the results and concepts of a particular topic have been put together. These are followed by a large quantity of solved examples. Quite a large number of problems have been given as exercises.I am thankful to Sundar Pathshala Prakashan Pvt. Ltd. for its contribution in bringing out this series in such a splendid form as well as thankful to staff who contributed in bringing out this series like as computer designer, art worker, etc. I also wish to thank all teachers and students who have given creative suggestions. I look forward to hearing from both teachers and students' opinions and valuable suggestion that will help improve the book in next edition.Author3rd Bhadra 2082Preface
Sets (Working hour 10) 71.1 Introduction of sets1.2 Definition of set1.3 Member of set 1.4 Method of writing set 1.5 Methods of describing set 1.6 \" Belongs to\" and \"Does not belong to\" 1.7 Mixed Exercise.Arithmetic (Working hour 45) 242.1 Whole neember 2.2 Integers 2.3 Fraction 2.4 Decimal 2.5 Percentage 2.6 Profit and loss 2.7 Unitary methodContent
Mensuration (Working hour 10) 1443.1 Distance 3.2 Perimeter 3.3 Area 3.4 VolumeAlgebra (Working hour 35) 1784.1 Indices 4.2 Algebraic Expression. 4.3 linear Equation and inequalityGeometry (Working hour 35) 2405.1 line and angles. 5.2 Triangle and Quadrilateral 5.3 Solids 5.4 Co-ordinates 5.5 Symmetry and TessellationsStatistics (Working hour 10) 3126.1 Collection of data6.2 Frequency Table
1UNIT SETS1.1 Introduction of SetsStudy the following information.It is a collection of flowers.It is a set of flowers.It is a collection of birds.It is a set of birds.It is a collection of even numbers less than 10.It is a set of even numbers less than 10.2 64 8It is a collection of fruits.It is a set of fruits.Mango belongs to set of fruits.Banana belongs to set of fruits.Acme Mathematics 6 7
Classwork1. Give the group name for each of the following collections:(a)o ae iu(b)5 71 39It is a collection of ................ It is a collection of ...............(c) (d)It is a collection of ................ It is a collection of ...............2. Cross the object which does not belong to the collection and write the name of the collection:(a) (b)It is a set of .................... It is a set of ...................(c) SundaySaturdayTuesdayJanuary(d)1 4 75 3 9It is a set of .................... It is a set of ...................8 Acme Mathematics 6
1.2 Definition of SetThe figure given below is a collection of flowers.It is a set of flowers. It is a set of numbers below 10.The figure given below is a collection of numbers less than 10.1837629 45Set is a well defined collection of discinct objects.It is a set of vehicle. Truck, Bus, Tractor, Motorcycle are the members of the set.There are 4 members.1.3 Member (Element) of a Set'a' is a member of the set.'a' belongs to the set.There are 5 members.e iaouAcme Mathematics 6 9
Look at the set of fruits.Litchee is a member of the set.Banana is a member of the set.Peach is a member of the set.Pear is a member of the set. Apple is a member of the set.The object of the set is its member. It is a set of fruitsClasswork1. Identify which of the following are sets.(a) Days of the week. (b) Months of the year.(c) Collection of birds. (d) Collection of fat boys.(e) Collection of thin girls. (f) Collection of musical instruments.(g) Collection of planets in the solar system.Kushal belongs to the set.Kushal is a member of the set.There are 5 members.Kushal ManjuKrishna SonikaKundan2. Write the name of the following sets.aoiueIt is a ........................It is a ........................It is a ........................10 Acme Mathematics 6
15 937It is a ........................It is a ........................1. Write 'True' or 'False' in the box.(a) Saturday is a member of 'Days of the week'.(b) 'a' is a member of 'English vowel sound'.(c) 'Parrot' is a member of 'Birds'.(d) 10 is a member of 'set of odd numbers'.(e) Orange is a member of 'set of fruits'.2. Name any four members of the following sets.(a) Musical instruments (b) Wild animals(c) Office furniture (d) Name of your teachers(e) Days of a week.3. Write 'T' for true and 'F' for false for each of the following statements about set.(a) Tuesday is a member of the set of days of a week.(b) Saturday is a member of the set of school running days of a week.(c) July belongs to the set of English months.(d) 3 is a member of the set of even numbers.(e) A triangle is not the member of the set of geometric shapes.(f) You are not a member of the set of your family.(g) The moon is a member of the set of planets.(h) Mathematics is a member of the set of books in class 6.Exercise 1.1Acme Mathematics 6 11
4. Write True or False in the blanks: (a) A school bag is a set of books, copies, pencils and eraser. (b) A instrument box is a set of compass, divider, set squares, protractor, pencil, and eraser. (c) A class is a set of students. (d) The collection of tall students is a set. (e) The collection of beautiful girls is a set. 1.4 Method of Writing SetWe can write the members of a set in many ways.Orange is a member of a set of fruits.Apple is a member of a set of fruits.Banana is a member of a set of fruits.Grapes is a member of a set of fruits.It is a set of fruits.We use capital letters A, B, C,..... to denote a set.for example :A = the set of fruitsB = {apple, orange, banana, grape}It is the set of seven days of the week.A = the set of seven days of the week.B = {Sunday, Monday,............., Saturday}In the set :(a) We write the members of a set inside the curly brackets { }.(b) We separate the members of a set by commas (,).(c) We use English capital letters to name the set.SundayTuesdayThursday FridaySaturdayMondayWednesday12 Acme Mathematics 6
Sets may be written by either of the following methods.A. By diagramIn this method member of the sets are circled. In the figure alongside 2, 4, 6, 8, 10 and 12 are the even members less than 13.B. By descriptionIn this method we describe the character of the members. For example : Sets given along side can be written as: The set of even numbers less than 13. The set of seven days of a week.C. By listingIn this method, we list the members inside the curly brackets. For example. A = {2, 4, 6, 8, 10, 12}S = {Sunday, Monday, Tuesday} Set of the letters in good = {g, o, d} Set of the letters in dog = {d, o, g}Note: In the set notation repetition of member is not valid. So, {g, o, o, d} is wrong in the set notation.1.5 Methods of Describing Sets(a) Description methodIn this method we describe the character of the members either in word or in sentence.For example: D = {the set of first 4 days of a week}(b) Listing method:In this method we write the member inside the curly bracket with comma.For example: D = {Sunday, Monday, Tuesday, Wednesday}(c) Set-builder method:In this method we write the member as variable (like x, y, z, etc.) and describe the common property of the variable.For example: D = {x:x is the first four days of a week}.2.13.6.8.4.10.8.10 .4 .12.6.2{ } is called curly bracket.Acme Mathematics 6 13
Solved ExampleExample 1 : Express A = { 2, 4, 6, 8, 10} in(a) Description method(b) Set-builder method.Solution: (a) A= {even numbers less than 12}(b) A = {x:x is a even number less than 12}Example 2 : Express B = {letters of the word 'word'} in listing method.Solution: B = {w,o, r, d}Classwork1. Write the members of the set using { } in your exercise book:(a) The set of the Devnagari numbers less than ten.(b) The set of the first 4 – days of a week.(c) The set of your books.(d) The set of your family members.(e) The set of fruits you ate yesterday.(f) The set of vehicles that you see.(g) The set of your teachers.2. Write the given set in description method:(a) A = {2, 4, 6, 8} (b) B = {Sunday, Saturday}(c) C = {apple, mango, orange} (d) D = {a, b, c, d}(e) E = {a, e, i, o, u} (f) F = {January, July, June}Exercise 1.21. Name and list the elements (members) of the following sets: (a) (b) (c) .10.11 .12 .13.14.15.5.20 .15 .10.25.z .y.x.v .w14 Acme Mathematics 6
(d) (e) (f)2. List the elements (members) of the following sets: (a) The set of first five months according to Nepali calendar. (b) The set of first four English months. (c) The set of days of week that begin with 'S'. (d) The set of first six letters of English alphabets. (e) The set of prime numbers less then 10. 3. Write the following sets by the description method: (a) { x, y, z} (b) {Tuesday, Thursday} (c) {1, 3, 5, 7, 9} (d) {+, –, ×, ÷}(e) {Red, Blue, Yellow, Orange, Black}4. Represent the following set in the diagram:(a) Days of a week (b) First 6 counting numbers(c) Our notes (d) Our coins(e) First 5 multiples of 10 (f) First 7 prime numbers.5. List the members of the following sets. [use listing method](a) F = the set of 5 fruits.(b) A = the set of first 3 months of the year. (use our calender)(c) N = the set of numbers less than 5.(d) T = the set of your uniform.(e) M = the set of 4 rivers of Nepal.(f) E = the set of colours of our national flag.Acme Mathematics 6 15
6. List the members of the given sets in the curly bracket.(a)5 1937(b)AsarBaishakhJesthaSawan(c)SantoshRamRamchandraHariSuman(d)TuesdayWednesdayThursdayMondaySunday7. Write the given sets in descriptive method. (Describing the members)(a) {1, 2, 3, 4, 5, 6, 7, 8, 9}(b) {Narayani, Bagmati, Koshi}(c) {Chitwan, Kaski, Ilam}(d) {Sun, Moon, Earth}(e) {a, e, i, o, u}(f) {Sagarmatha}8. A = {factor of 18} is a given set. Answer the following questions.(a) Rewrite the set in listing method.(b) Is 9 a member of the set A? Give reason.(c) Express the set in set builder form.Remember!Q. No. 116 Acme Mathematics 6
Consider the set A = { a, b, c, d}.Here, a, b, c, and d are the elements of the set A. Thus member of a set is called element of a set. We read A and a as; 'a belong to A' i.e. 'a' is the member of the set A.'a' is the element of the set A. It is written as, 'a'∈A.Thus, b∈A, c∈A, d∈ABut e∉A'∉' indicates 'does not belong to.'e' is not a member (element) of the set A.Classwork1. Cross () the odd and write the name of the sets of the remaining members.(a) (b)(c) (d)(e) {5, 10, 15, 20, 31} (f) 76, 54, 43, 12I know ∈ is a Greek letter. ∈ is epsilon.1.6 'belongs to' and 'does not belong to'.Acme Mathematics 6 17
2. Write 'True' or 'False' in the blank.(a) Orange ∈{fruits of Nepal} .................................(b) 5 ∉{prime number} .................................(c) Narayani ∈ {river of Nepal} ................................(d) Father ∉ {member of family} ................................(e) Green ∈ {traffic light} ..................................Exercise 1.31. Write True or False in the blanks:(a) Ram ∈ the set of days in a week. (b) 2 ∈ the set of even numbers.(c) 2 ∈ the set of prime numbers. (d) Rat ∈ set of fruits.(e) Tiger ∈ the set of domestic animals.(f) March ∈ {Sunday, Monday, Tuesday}(g) 6 ∈ {1, 2, 3, 4, 5}2. If R ={r, a, t} and S ={s, o, n} are two sets, put the symbols ∈ or ∉ in the blanks.(a) r ............ R (b) w ............. R (c) s ........... S(d) o ............ S (e) t .............. S (f) n ............ R3. Choose the well-defined statements.(a) The smallest students. (b) Days of a week.(c) Districts of Bagmati zone. (d) Our coins. (e) The best teacher.4. Write 'True' or 'False' in the blanks.(a) Set is a collection of things .................(b) Set is a collection of well-defined objects ................. (c) Set is denoted by A, B, C etc. .........................(d) Set is denoted by a, b, c etc..............................(e) ∈ is symbol for 'does not belong to' ..............(f) Member of the set is denoted by a, b, c, etc. ................(g) The order of the members in a set has no meaning............... 18 Acme Mathematics 6
1. Identify which of the following are sets.(a) Days of the week. (b) Months of the year.(c) Collection of birds. (d) Collection of fat boys.(e) Collection of thin girls. (f) Collection of musical instruments.(g) Collection of planets in the solar system.2. List any two elements in the blanks.(a) {Districts of Nepal} ................ (b) {Days of a week} ...............(c) {Even number less than 9} ............ (d) {Our Nepali months} ..........(e) {Our coins}.............................3. Write in description method.(a) {1, 3, 5, 7} (b) {2, 4, 6, 8,10} (c) {cow, dog}(d) {Baishakh, Jestha} (e) {Sunday, Saturday} (f) {The set of whole number less than 10}4. Write the following sets in set-builder form.(a) {a, b, c, d} (b) {x, y, z} (c) { a, e, i, o, u}(d) {5, 10, 15, 20} (e) {January, June, July} 5. Write the following sets in listing method.(a) A= {x:x is odd number less than 12}(b) X ={x:x is prime number less than 10}(c) Z = {x:x is natural number between 2 and 7} (d) B ={x:x is colours of traffic light}(e) C = {x:x is notes of Nepalese currency in use}6. Even number between 1 and 20 are given.2, 4, 6, 8, 10, 12, 14, 16, 18(a) State whether the given numbers are well-defined or not? Give reason.(b) Write given numbers in set notation.(c) Write the given set in 'description method'.Mixed ExerciseAcme Mathematics 6 19
7. Two sets are, A = {pen, pencil, eraser, compass} and B = {book, pen, protractor, divider}(a) Inter the proper symbol (∈ or ∉)(i) Book .......... set A. (ii) Eraser ......... set B.(iii) Compass ......... set A. (iv) Divider ........... set B.(b) Write the number of elements in sets A and B.(c) How many elements are same in sets A and B?8. Study the given figure and answer the question given below.(a) How many elements are there?(b) Write set Q by using listing method.(c) Write set Q by set builder method.9. Write the following as indicated below.(a) Set A = {2, 4, 6, 8} by describing method.(b) Set B = {English vowel sound} by set-builder method.(c) Set C = {x : x is a prime number less than 10} by listing method.10. D denotes odd number less than 16.(a) Write all members of set D.(b) Write the total number of member of set D.(c) Is '10' a member of set D?11. Observe the given informationSunday, Monday, Tuesday, Wednesday, Thursday(a) Is the above information well-defined?(b) Write the information in set notation.(c) Write the information in diagram.12. Answer the following questions based on the given collected data: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday(a) State whether the given days is well-defined or not? Write with reason.(b) Write the days in set notation.(c) Write the given set in a set-builder method.13. Study the given figure and answer the question given below.(a) Write the number of members of set 'A'.aieuoQ.2.6.8.4.10A20 Acme Mathematics 6
(b) Write set 'A' by using listing method.(c) Write set 'A' by using describing method.14. Write the following as indicated below.(a) A = {3, 6, 9, 12, 15} by describing method.(b) T = {The set of composite numbers less than 10} (by set-builder method)(c) R = {x/x is a multiple of 4 less than 40} (by listing method.)15. Answer the following questions based on the given numbers.4, 6, 8, 9, 10, 12, 14, 15, 16, 18(a) State whether the given collection of number is well defined or not? Write with reason.(b) Write the numbers in set notation.(c) Write the given set in a set builder method.16. If X denotes even numbers less than 15,(a) Write all members of set X by listing method.(b) Write the number of members in set X.(c) Is '8' a member of given set X?17. Study the collection given below and answer the following questions.Pig, Goat, Crow, Cow, Sheep(a) Write the number of elements of given collection.(b) Find odd one out from the given collection with a suitable reason.(c) Write the name of the set formed after removing odd element.18. X = {pen, pencil, eraser, compass}, Y = {book, pen, protractor, compass}(a) Insert the proper symbol [belongs to (∈) or does not belong to (∉)]i. Book ................... set Y ii. Eraser .................. set Xiii. Compass .............. set X iv. Protractor ............ set X.(b) Find number of members in set A and number of members in set B.(c) Are the sets X and Y have equal number of members?Acme Mathematics 6 21
Project Work1. Objective : To make different sets.2. Materials required : A-4 Size Paper Pen3. Activity : List the things in your home. Classify them in the group. Name the group and present it to your class room.For example : Kitchen items : ..............................................................................................................................................................................................................................22 Acme Mathematics 6
EvaluationTime: 30 minutes Full Marks: 121. (a) Answer the following questions based on the given collected numbers. 2, 4, 6, 8, 10 (i) State whether the given collection of numbers is well–defined or not? Write with reason. [1] (ii) Write the given set in a describing method. [1] (b) If set A={1,2,3,4} and Set B={a,b,c,d}, put ∈ or ∉ in the blanks. [1] (i) 3 ........... A (ii) e .............B2. ‘X’ denotes even numbers less than 15. (a) Write all members of X by listing method. [1](b) Write the number of elements of set X. [1](c) Is '8' a member of given set X ? [1]5. (a) Define set with example. [1](b) If, V = {Vowel letter of English alphabet} write the members of set 'V' in listing method. [1](c) Is m∈V ? Give reason. [1]6. Study the collection given below and answer the following questions. Pig, Goat, Crow, Cow, Sheep (a) Write the number of elements of given collections. [1](b) Find odd one out from the given collection with a suitable reason. [1](c) Write the name of the set formed after removing odd element. [1]Acme Mathematics 6 23
2UNIT ArithmeticsWarm Up Test1. Write ten Hindu-Arabic digits.2. Write ten digits of Devnagari number system.3. Write the place value of the circled digits.(a) 2 34 (b) 546 7 (c) 34 5 67 (d) 2 345674. Write in national place value chart: 1097345. Write in international place value chart: 932076 6. Use comma (use both systems)(a) 73296 (b) 8034037. Write in power of 10.(a) Lakh (b) Crore (c) Kharab(d) Thousand (e) Million (f) Billion 8. Write the number name: 1020304056789. Write the number: Three kharab two arab10. Fill in the blanks.Number of digits Smallest number Largest number71011611. Simplify:(a) 13 + (19 - 10) – 7 (b) 100 + { 2 + (9 - 7) + 2} + 1012. Choose the numbers divisible by 2, 3 or 5.82104312355931532308107024 Acme Mathematics 6
13. Write in number (use international number system)(a) Five million two hundred thirty seven thousand three hundred and forty.(b) Nine hundred twenty billion seven hundred ten million eight hundred fifty five thousand six hundred twelve.14. Round the numbers.(a) 567 (nearest hundred) (b) 6789 (nearest ten)(c) 12346 (nearest thousand)15. cIf/df n]Vg'xf];\\M(a) #$%^&*( (b) $))%))*( (c) &*())&*)*16. Check divisibility by 2, 3, 5 and 10.(a) 700 (b) 819 (c) 8750 (d) 246817. Write all factors of:(a) 25 (b) 80 (c) 10018. Find the prime factors:(a) 102 (b) 140 (c) 25219. Calculate the HCF of the given numbers.(a) 7 and 12 (b) 64 and 72 (c) 50 and 68 (d) 36 and 4520. Calculate the LCM of the given numbers.(a) 6 and 8 (b) 10 and 12 (c) 75 and 10021. Find the square roots of the following numbers.(a) 196 (b) 625 (c) 101422. Find the cube roots of the following numbers.(a) 27 (b) 125 (c) 729(d) 343 (e) 216Acme Mathematics 6 25
2.1 Whole NumberA. Development of whole numbersStudy the following:I read in class - 6I read in class - VII read in class - Sixd ^ sIffdf k9\\5' . Sushma Rohan Puja SamirSushma, Rohan, Puja and Samir all are in class six.6, VI, ^ and six are the symbols for the number.Counting is as old as history of human civilization. Different civilizations followed different methods of counting by using different symbols.The different symbols are:0, 1, 2, 3, 4, 5, 6, 7, 8, 9 …… Hindu-Arabic numbersI, II, III, IV, V, VI, VIII , ….. Roman numbers), !, @, #, $, %, ^ =============== Devnagari numbers1, 2, 3, 4, are called numerals (digits). Numerals represents the numbers.The counting numbers 1, 2, 3, 4, ….... etc. are called natural numbers. It is denoted by N,N = {1, 2, 3, 4, 5, …….}1 2 3 4 5 6 7 81 is the smallest natural number� There is no end of natural number� There is no largest natural numberWhat about 0 ?The number 0 together with set of Natural number are called Whole numbers. It is denoted by W.W = {0, 1, 2, 3, 4, 5, 6,…...}0 1 2 3 4 5 6 7 8 90 is the smallest whole number.Zero 0 ?26 Acme Mathematics 6
� There is no end of whole number. � There is no largest whole number.0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are ten digits, so this system is also called decimal system or base 10 system.(a) Place value and face value:Face value: The face value of a digit is same where ever it lies. For example the face value of 2 in 12 or 24 or 234 or 20143 is 2 only.Place value: The place value of a digit is different according to its place. For example the place value of 2 in 12 is 2.24 is 20 (twenty)234 is 200 (two hundred)20143 is 20000 (twenty thousands)(b) Place value chartNational system (Our system)Places Ten Kharabs Kharab Ten arabs Arab Ten Crores Crore Ten Lakhs Lakh Ten Thousands Thousand Hundreds Tens Ones8 9 0 9 8 7 6 5 4 3 2 1 0The numbers is 8909876543210.It is 13 digits numbers.Comma is used to separate the periods. Same periods are read together. 89, 09, 87, 65, 43, 210Eighty nine kharab nine arab eighty seven crore sixty five lakh forty three thousand two hundred ten.International systemPeriods Billions Millions Thousand UnitsPlaces Hundred Billions Ten Billions Billions Hundred Millions Ten Millions Million Hundred Thousand Ten Thousands Thousand Hundred Tens Ones9 0 9 8 7 6 5 4 3 2 1 0Remember! The rule to put comma now.Acme Mathematics 6 27
909, 876, 543, 210 ⇒ Nine hundred nine billion eight hundred seventy six million five hundred forty three thousand two hundred ten.(c) Number and number of digits1, 2, 3,…………….... 9 ⇒ one digit numbers10, 11,……………..... 99 ⇒ two digit numbers100, 101,…………..... 999 ⇒ three digit numbers1000, 1001, ……....... 9999 ⇒ four digit numbersBy using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 we can make any number either large or small.Solved ExampleExample 1 : Find the greatest and smallest number formed by 0, 1, 2, 3.Solution: For the greatest number we arrange the digits in decreasing order. So, greatest number is 3210.For the smallest number we arrange the digits in increasing order, so the smallest number is 0123. Since 0 has no meaning here, So, we can write 1023.Example 2 : Find the difference in the place value of the two 8s in 38275820.Solution: The 8s are in the hundred and ten lakhs places.Now, place value of 8 in the hundred place = 800 Place value of 8 in ten lakh place = 8000000. Difference = 8000000 – 800 = 7999200Example 3 : How many 2-digit numbers are there in all ?Solution: The largest 2-digit number = 99The largest 1 digit number = 9Total number of 2-digit numbers = 99 – 9 = 90Classwork1. Fill in the blanks.(a) 1 lakh = …………… ten thousands (b) 10 lakhs = ……… million1 hundred = ………….. tens 1 crore = ……… million 100 thousand = …… lakhs 2 billion = ……… lakhs'0' is place holderIt is the smallest 4-digits numberIt is the largest 4-digits number28 Acme Mathematics 6
1 crore = …….. lakhs 10 billion = ……… crore2. How many 3-digits numbers are there in all ?3. How many times does the digit 2 occur in the number between 2 and 100 ?4. Is every natural number \"a whole number\" ? 5. Is every whole number \"a natural number\" ?Exercise 2 .11. Put in the place value chart and write the place value of the circled digits.(a) 12 3 40789 (b) 9 8 7062563409(c) 10000 5 00003000 (d) 1203 4 056078092. Find the greatest and smallest number formed by the digits 2, 5, 7, 9, 6 and find their sum also.3. Find the difference between the values of the two 5s in each numbers. (a) 3456785 (b) 52900577 (c) 5007250072 (d) 85432095004. In the natural number system;(a) How many 1-digit numbers are there in all ?(b) How many 2-digit numbers are there in all ?(c) How many 4-digit numbers are there in all ?5. Find the sum of the greatest and the smallest numbers of 9 digits. 6. Write the number names in both systems using comma.(a) 1230067891213 (b) 998866725678(c) 700005000600 (d) 8792654012347. Write 'True' or 'False'.(a) 0 is a natural number. (b) 8 is the greatest 1-digit number.(c) 10 million makes a crore. (d) 1 crore makes 1000 thousand. (e) 10 is not a whole number. (f) Lakh makes 10000 hundred.(g) The smallest 9-digit number is 111111111.8. Do the following:(a) Find the sum of the smallest and largest number of 4-digit.(b) Find the sum of the numbers that are smallest of 4-digit number and largest of 6-digit number.(c) Calculate the sum and difference between the place values of two 8's in the number: 80678921.(d) Calculate the sum and difference between the face value and place value of 5 in the number : 987506000.Acme Mathematics 6 29
B. Simplification with BracketsActivityStudy the given statement carefully.I went to Birat Stationery and buy the followings 2 pens of each cost Rs. 253 erasers of each cost Rs. 31 note copy cost Rs. 35I have Rs. 100.How much money is left with me ? This can be solved as: 100 – (2 × 25 + 3 × 3 + 1 × 35)= 100 – (50 + 9 + 35) = 100 – 94 = 6Rs. 6 is left with me.This process of solving the problem is called simplification. Simplification has its own rule. Simplification contains the sign +, –, ×, and ÷. Simplification contain the brackets ( ), { }, and [ ]. Rule of sign in the order is D (divide), M (multiply), A (addition), and S(subtraction). In short it is DMAS. Rule for bracket is ( ), { }, and [ ] or [ { ( ) }] Solved ExampleExample 1 : Simplify: 24 ÷ 3 + 3 × 5 – 6Solution: Here, 24 ÷ 3 + 3 × 5 – 6= 8 + 3 × 5 – 6 Division is done.= 8 + 15 – 6 Multiplication is done.= 23 – 6 Addition is done.= 17 Subtraction is done'of' is used as multiplication sign.First in operationSecond in operationThird in operationDMAS is used30 Acme Mathematics 6
Example 2 : Simplify: 70 + [50 - {6 + (16 - 7)}]Solution: Here, 70 + [50 – {6 + (16 – 7)}]= 70 + [50 – {6 + 9}]= 70 + [50 – 15]= 70 + 35= 105Example 3 : Simplify: {27 + 9 (6 – 4) – 9}Solution: Here, {27 + 9 (6 – 4) – 9}= {27 + 9 × 2 – 9}= {27 + 18 – 9}= {45 – 9}= 36Example 4 : Simplify: 5 of 3 + 2 (30 ÷ 5)Solution: Here, 5 of 3 + 2 (30 ÷ 5)= 15 + 2(30 ÷ 5)= 15 + 2 × 6= 15 + 12 = 27Example 5 : Simplify: [6 × {5 + (6 – 1 + 2) ÷ 3}]Solution: Here, [6 × {5 + (6 – 1 + 2) ÷ 3}] = [6 × {5 + (6 – 3) ÷ 3}]= [ 6 × {5 + 3 ÷ 3}]= [ 6 × {5 + 1}]= 6 × 6 = 36Example 6 : Find the sum of money when 3 students bought one pen for Rs. 30 and one eraser for Rs. 5 each.Solution: Cost of pen and eraser each = Rs. 30 + Rs. 5Cost for 3 students = 3 (30 + 5)Now, sum of money = 3(30 + 5) = 3 × 35 = 105Thus, the sum is Rs. 105Subtraction inside the ( ) is first operation.Addition inside the { } is second operation. Subtraction inside [ ] is 3rd operation.We put '×' sign if there is no sign between a number and bracket.It is vinculum sign.Its operation is done at firstAcme Mathematics 6 31
The result of simplification depends upon the position of brackets.Simplify: (9 + 2) × 3 Here, (9 + 2) × 3 = 11 × 3 = 33 Simplify: 9 + 2 × 3 Here, 9 + 2 × 3 = 9 + 6 = 15 Now, compare the resultOrder of Brackets : The order of brackets is given below.( ) → round bracket, First Step{ } → curly bracket, Second Step[ ] → big bracket, Third Step[ { ( ) } ]The simplification inside the brackets also follows the order D – M – A – S.Example 7 : Find the result when the difference between 20 and 12 is multiplied by 14 and divided by 7.Solution: The mathematical statement is : {14 × (20 – 12)} ÷ 7Now, {14 × (20 – 12)} ÷ 7= {14 × 8} ÷ 7= 112 ÷ 7= 16Example 8 : Write the order of rules when we do simplification.Solution: If +, –, ×, +, of, ( – ), { }, ( ), [ ] includes in the simplification, we follow the order as:B – brackets O – of D – divisionM – multiplication A – addition S – subtractionNote: vinculum sign (or bar) is operated first, it is also a kind of bracket.Classwork1. Simplify:(a) 12 + (7 – 4) (b) (18 – 12) ÷ 2(c) 5 + (5 × 3) ÷ 3 (d) 200 – (40 ÷ 8) – 100(e) 30 – (7 + 4) ÷ 11 (f) 73 + (4 + 3) × 6(g) 18 × (2 + 6) ÷ 4 (h) (26 × 3) + 8 – 1032 Acme Mathematics 6
2. Put the small bracket in the appropriate place so that the answer is correct:(a) 100 + 50 ÷ 10 × 5 (b) 100 + 50 ÷ 10 × 5 (c) 100 + 50 ÷ 10 × 5Ans : 101 Ans : 125 Ans : 75Exercise 2.21. Fill in the blanks.(a) 2 × 3 – 4 = .................... (b) 16 + 4 – 2 = .................(c) 15 + 5 + 4 = .................... (d) 4 × 2 + (6 – 4) = .................(e) 20 + 2 × 2 + 2 = .................... (f) 4 + 60 – 6 = .................2. Simplify:(a) 4 + (5 – 2) (b) (10 – 5) + 3 (c) 1 – (6 + 12)(d) 10 ÷ (2 + 3) (e) 44 ÷ (4 + 7) (f) 100 ÷ (40 – 20)(g) 20 ÷ (2 × 5) (h) (12 × 4) ÷ 2 (i) (40 + 20) ÷ 4(j) (6 + 3) × 2 (k) 25 ÷ (2 + 3) × 4 (l) 12 ÷ (8 – 2) × 103. Make the mathematical expression and write in the blanks.(a) The sum of 2 and 10 = ...............................(b) 15 times the sum of 2 and 7 = ...............................(c) The sum of 13 and 7 divided by 4 = ...............................(d) 20 times the difference of 5 and 2 = ...............................(e) The quotient of 30 divided by the difference of 8 and 3 = ...................4. Simplify:(a) 44 – 5 × 8 + 6 ÷ 2 (b) (24 – 5) × (8 + 6 ÷ 2)(c) 24 – (5 × 8 + 6) ÷ 2 (d) (24 – 12) ÷ (6 – 3)(e) {18 ÷ (5 – 2) + 2} (f) 13 – 8 ÷ 4 – 1(g) (21÷ 3) (10 – 3) (h) 24 ÷ (12 – 4) × 35. Simplify:(a) 36 ÷ {(13 + 2) – (9 – 6)} + 6 (b) 20 + (8÷ 4) + 8 × {(16 – 12) ÷ 4}(c) [{30 ÷ (1 + 5)} + 12] ÷ 17 – 10 (d) 56 + 2[231÷ {3 + (7 – 2 + 3)}](e) 39 – [4{16 ÷ (5 – 1)} – 3] (f) 14 + [8 + {4 + 7 – 2 – 8}](g) [99 ÷ 9 + {(14 – 16 + 5 – 1)}] (h) 32 + [12 + {16 ÷ 8 – (12 – 3 of 2)}](i) [100 – (18 – 2) ÷ 4 + 10] of 7 – 20Acme Mathematics 6 33
6. Write the mathematical sentences for the following statements and simplify:(a) 10 is added to the product of 2 and 3.(b) The product of 2 and 7 is added to the difference of 12 and 5. (c) 7 times the difference of 19 and 17 is divided by 14.(d) 45 is divided by 9 and the quotient is added to the difference of 20 and 4.(e) 18 is added to the three times of the quotient of the sum of 60 and 70 divided by 13.(f) The difference of 20 and 14 is added to 3 times the sum of 9 and 7.(g) Take a number 15, double it, subtract 14 and divide the result by 8.C. Test of divisibilityWe all know that division is one of the most important operation in mathematics. It has own rules which help us to find the solution of the given problems.Now, study the given division carefully12)1367(113– 1216– 1247– 3611QuotientDividendDivisorRemainderJivan, here 12 and 1367 are divisor and dividend respectively.Yes Gitu, we know that 113 is quotient and 11 is remainder.Hence, The number to be divided is the dividend. The number which divides is the divisor. The remaining number after dividing is the quotient. The number left over from the dividend is the remainder.The rules which are used to find out whether the number is a divisor of a given number or not is called the test of divisibility.The rules of division help us to check whether a number is a divisor of a given number or not without actually performing the division. This rule is called the test of divisibility. Lets study the rules.(a) Divisibility by 2A number is divisible by 2, if the last digit of the number is 0, 2, 4, 6, or 8 23408 and 62900 are divisible by 2.23407 and 62913 are not divisible by 2. Why ?34 Acme Mathematics 6
(b) Divisibility by 3A number is divisible by 3, if the sum of the digits of the number is divisible by 3. 23418 is divisible by 3, since 2 + 3 + 4 + 1 + 8 = 18, divisible by 3.23428 and 2911 are not divisible by 3(c) Divisibility by 5A number is divisible by 5, if the last digit of the number is 0 or 5. 2300 and 6705 are divisible by 5.2303 and 6704 are not divisible by 5.(d) Divisibility by 7A number is divisible by 7, when we multiply the last digit by 2 and subtract the answer from the remaining number, if the result is 0 or divisible by 7, the original number is divisible by 7.Consider the number 392.Here, the last digit = 2. So, 2 × 2 = 4 Remaining number = 39Difference = 39 − 4 = 35.35 is divisible by 7.Hence, 392 is divisible by 7. Similarly, 17976 is divisible by 7.But, 50144 and 17975 are not divisible by 7.(f) Divisibility by 10A number is divisible by 10, if the last digit of the number is 0. 50 and 100 are divisible by 10.105 and 3005 are not divisible by 10.(f) Divisibility by 11A number is divisible by 11, when the difference between the sum of every digits in the odd place and even places of that number is 0 or 11.Consider the number 50171.Here, Digits on the odd places are 1, 1, and 5 Digits on the even places are 7, and 0 Difference = (1 + 1 + 5) − (7 + 0) = 0.Hence, 50171 is divisible by 11.Similarly, 538516 is divisible by 11.But, 56171 and 528516 are not divisible by 11.Acme Mathematics 6 35
Classwork1. Write two 3-digit numbers that are divisible by the following numbers.(a) 2 → 234 and 302 (b) 3 → ..... and .....(c) 7 → ..... and ..... (d) 5 → ..... and .....(e) 11 → ..... and .....2. Colour the following numbers by using the given colour code:Number exactly divisible by 2 redNumber exactly divisible by 3 yellow14 21 99 220 123 22428 405 506 513 225 1003. Check and tick () divisibility by 2, 3, 4, and 5:Number 2 3 4 5244090105222Exercise 2.31. State whether the following sentences are true or false.(a) 1234 is divisible by 2 (b) 4318 is divisible by 2 and 3(c) 56070 is divisible by 5 (d) 7893 is divisible by 3(e) 7000 is divisible by 5 2. Which of these numbers are divisible by 2?(a) 439 (b) 555 (c) 600 (d) 98736 Acme Mathematics 6
3. Which of these numbers are divisible by 3?(a) 423 (b) 93 (c) 104 (d) 1534. Which of these numbers are divisible by 5?(a) 80 (b) 96 (c) 335 (d) 1045. Which of these numbers are divisible by 10?(a) 100 (b) 215 (c) 310 (d) 4556. Colour the following numbers by using the given colour code:Number exactly divisible by 4 redNumber exactly divisible by 5 yellow24 50 96 324 75 112315 105 210 445 408 2447. Check the divisibility and tick the box if it is divisible and cross if it is not divisible.Numbers 2 3 5 7 111320120067203080302408. Fill in the row 2 of the table with the smallest digit in the blank which will make it divisible by the number in the row 1.Divisor (row - 1) 2 3 7 11Number (row - 2) 231 ... ... 237 9987 ... 8 ... 9484Acme Mathematics 6 37
D. Multiples and factorsStudy the following table carefully.Multiplication table of 2 = 2, 4, 6, 8, 10, 12, 14, 16Multiplication table of 4 = 4, 8, 12, 16 , 20, 24 Multiplication table of 8 = 8, 16 , 24, 32, 40Multiplication table of 16 = 16 , 32, 48, 6416 is divisible by 1, 2, 4, 8 and 16So, all these numbers 1, 2, 4, 8 and 16 are factors of 16.16 is a multiple of all these number 1, 2, 4, 8 and 16, because 16 comes in the multiplication table of these numbers.Look at the another table given below carefully.Factors Multiple1 × 302 × 153 × 105 × 630303030 The first column shows two factors multiplied together The second column shows the product of the two factors.Look at this table:2 × 1 = 2 2 × 4 = 8 2 × 2 = 4 2 × 5 = 10 2 × 3 = 6 2 × 6 = 12Here, 2, 4, 6, 8, 10 and 12 are divisible by 2.Similarly, 4, 8 and 12 are divisible by 4.6 and 12 are divisible by 6.126 410 282 12 484 12 6 6Since 2 × 15 = 3030 ÷ 2 = 1530 ÷ 15 = 2It is division fact.Oh! Multiple of 5 are 5, 10, 15, 20,..38 Acme Mathematics 6
We can write, 5 as 1 × 510 as 2 × 515 as 3 × 5E. Prime and Composite numbersLook at the following numbers carefully:Group-A Group-B2 = 2 × 1 4 = 2 × 2 × 13 = 3 × 1 6 = 2 × 3 × 15 = 5 × 1 8 = 2 × 2 × 2 × 17 = 7 × 1 9 = 3 × 3 × 111 = 11 × 1 10 = 2 × 5 × 113 = 13 × 1 12 = 2 × 2 × 3 × 114 = 2 × 7 × 1In a group-A, Every number has only two factors 1 and the number itself. These are prime numbers. In group-B, Every number has more than two factors. These are composite numbers. Thus, prime number are those numbers which has only two factor 1 and the number itself.Composite numbers are those numbers which has more than two factors.2, 3, 5, 7, 11 and 13 are the prime numbers less than 15.4, 6, 8, 9, 10, 12 and 14 are the composite numbers less than 15.102015 530 2535 405It is multiple of 5.3 and 5 are factors of 151, 2, 4,8, 1616FactorsMultipleI know it. 1 is a factor of every number. Any number is factor of itself.Acme Mathematics 6 39
2 is only even prime number.I know 1 is neither prime nor composite number.Study the given table carefully.Number Factors Number of factors1 1 12 1, 2 23 1, 3 24 1, 2, 4 35 1, 5 26 1, 2, 3, 6 47 1, 7 28 1, 2, 4, 8 49 1, 3, 9 310 1, 2, 5, 10 411 1, 11 212 1, 2, 3, 4, 6, 12 6Now, answer these questions(a) How many numbers have only one factor ?(b) How many numbers have only two factors ?(c) How many numbers have more than two factors ?The numbers 2, 3, 5, 7,………..which have only two factors are called prime numbers.The numbers 4, 6, 8, 9, 10, 12, …… which have more than two factors are called composite numbers. The number 1 is neither prime nor composite number. 2 is only even prime number. All other prime numbers are odd.40 Acme Mathematics 6
F. Methods of Finding Factors(a) Finding all factorsThere are two methods to find the factors of a given number. They are:(i) Multiplication method(ii) Division methodLet us learn these methods by examples.Example 1 : Find all the factors of 12. Solution:(i) Multiplication method:We have,4 × 3 = 126 × 2 = 1212 × 1 = 12So, 1, 2, 3, 4, 6 and 12 are factors of 12.(ii) Division methods:We have,12 ÷ 1 = 1212 ÷ 2 = 612 ÷ 3 = 412 ÷ 4 = 312 ÷ 6 = 212 ÷ 12 = 1So, 1, 2, 3, 4, 6, and 12 are factors of 12.(b) Finding prime factorsFactors of a number which are prime are called its prime factors. For example :Factors of 24 are : 1, 2, 3, 4, 6, 8, 12, 24Prime factors of 24 are: 2 and 3 only24 can be written as : 2 × 2 × 2 × 3A process in which every factor is prime is called prime factorisation. For examples :20 = 2 × 2 × 5, 27 = 3 × 3 × 3, 50 = 2 × 5 × 5There are two methods to find the prime factors of a given number. They are:(i) Division method (ii) Factor tree methodI understood! The quotient and divisor both are factors.Acme Mathematics 6 41
(i) Division methodPrime factors of a number can be find by repeated division by prime number.Example 1 : Find the prime factors of 30.Solution: Divide 30 by the first prime number 2.Divide 15 by the prime number 3.The quotient is 5 which is also prime.As a product of its prime factors, 30 can be written as,30 = 2 × 3 × 5Hence, 2, 3 and 5 are prime factors of 30.Example 2 : Write 60 as product of prime numbers.Solution: Here,2 60 → dividing by 22 30 → dividing by 23 15 → dividing by 35 → It is prime numberSo, 60 = 2 × 2 × 3 × 5(ii) Factor tree methodPrime factors of a number can also be found by factorizing in pictorial form, which is called a factor tree.Example 3 : Find the prime factors of 70.Solution: Look at the process carefully.Step-1 Step-2702 3570 = 2 × 35 702 355 770 = 2 × 5 × 730155232 705 35742 Acme Mathematics 6
702 355 7In this process we continue factorising until all factors are prime.The shape is like tree so, it is factor tree.Note : We can form the factor tree in another way also.Example 4 : Find all the prime factors of 12.Solution:(i) Division method Here,2, 2 and 3 are prime numbers.So, prime factors of 12 are 2, 2 and 3.(ii) Factor tree method. Here.2, 2 and 3 are prime numbers So, prime factors of 12 are 2, 2 and 3.The process of expressing a given number as a product of prime factors is called prime factorization.(c) Finding multiplesConsider the number 3 and set of natural number N = {1, 2, 3, 4, ....} Now, multiplying the number 3 by the natural number. We have,3 × 1 = 33 × 2 = 63 × 3 = 93 × 4 = 12 Here, 3, 6, 9, 12, are multiples of 3.Multiple of any number is the product of natural number and the given number.2 122 63122 × 32 × 6Multiples are infinite but factors are finite.Acme Mathematics 6 43
Classwork1. Colour the numbers orange that have only two factors:28 19 23 462 6 5 10 11602. Colour the numbers yellow that have more than two factors:41 55 65 477 9 21 23 40613. Write all the factors of the given numbers:(a) 6 (b) 15 (c) 34 (d) 44(e) 20 (f) 28 (g) 49 (h) 554. Write the given numbers as the product of prime factors:(a) 18 (b) 20 (c) 30 (d) 46(e) 56 (f) 72 (g) 81 (h) 92(i) 64 (j) 12 (k) 28 (l) 99Exercise 2.41. Fill in the blanks.(a) Prime numbers between 1 and 10 are ........... .(b) ........... is the only even prime number.(c) ........... is neither prime nor composite number.(d) ........... are the composite numbers between 10 and 20. (e) Factors of 6 are …………. (f) Factors of 15 are ………….(g) Prime factors of 30 are …………. (h) Prime factors of 42 are ……….(i) First 3 multiples of 2 are …………. (j) First 4 multiples of 5 are ....……2. Fill in the blanks. 'True or False':(a) Every number is multiple of itself ...........(b) 2 is an even prime number. ................(c) 1 is the smallest prime number ............(d) All even numbers are multiple of 2 ........44 Acme Mathematics 6
3. List the prime and composite numbers from the given numbers:515562574739295918 24 11 6 250494. Write the number from 51 to 80 and colour the prime numbers by red and composite numbers by green:51 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 805. List the following elements(a) All the factors of 10, F10 (b) All the factors of 15, F15(c) All the factors of 16, F16 (d) First 3 multiples of 6, M6.(e) First 4 multiples of 7, M7. (f) First 3 multiples of 10, M10.(g) Prime numbers between 1 and 15, P15. (h) Prime numbers between 90 and 100, P100. (i) Composite numbers between 15 and 25, C25.(j) Composite numbers between 90 and 100, C100. 6. Write the number from 25 to 100 and list the following.(a) set of even numbers, E (b) Set of odd numbers, O(c) Set of prime numbers, P (d) Set of composite numbers, C.7. Complete the factor tree.(a) (b) (c)(d) (e) (f)362 18453 15502 25982 491143 381255 25Acme Mathematics 6 45
(g) (h) (i)8. Express the following numbers as the product of the prime number. (Use factor tree method)(a) 120 (b) 630 (c) 588 (d) 800 (e) 8259. Find all the factors of the given numbers.(a) 15 (b) 25 (c) 30 (d) 42 (e) 4410. Express the following numbers as the product of the prime factors.(a) 18 (b) 24 (c) 75 (d) 90 (e) 210(f) 275 (g) 625 (h) 150 (i) 540 (j) 94511. Find the product of common factors of the given numbers.(a) 18 and 24 (b) 20 and 32 (c) 48 and 72(d) 27 and 36 (e) 24 and 60 (f) 150 and 12012. Find the smallest common multiple of the given numbers.(a) 2 and 3 (b) 4 and 6 (c) 5 and 15(d) 6 and 8 (e) 8 and 12 (f) 9 and 15Project WorkWrite the numbers from 1 to 100 and make a chart and do the following activities. (a) colour the numbers red which are divisible by 2. (b) colour the numbers green which are divisible by 3. (c) colour the numbers yellow which are divisible by 5. (d) colour the numbers orange which are divisible by 7. After completing the colouring, present it to your class room.6×16×××30××46 Acme Mathematics 6
G. Highest Common Factor (HCF) and Lowest Common Multiple (LCM)A. Common factorMany numbers can divide many numbers, without remainder. For example:2 can divide 4. 2 can divide 6. 2 can divide 8.2 is a divisor of 4, 6 and 8. So, 2 is a factor of 4, 6 and 8. 2 is a common factor of 4, 6 and 8.Example: Find the common factors of 8 and 6.Solution: Factors of 8 = 1, 2, 4 and 8.Factors of 6 = 1, 2, 3 and 6.Common factors of 8 and 6 = 1 and 2.B. Highest common factorConsider the numbers 12 and 18.Factors of 12 = 1, 2, 3, 4, 6 and 12.Factors of 18 = 1, 2, 3, 6, 9 and 18.Common factors = 1, 2, 3, and 6.Highest common factors = 6Thus, HCF is 6.The highest common factor (H.C.F) is the greatest number which divides two or more numbers without remainder. We use the following methods to find the HCF.(i) Set of all factors (ii) Prime factorization (iii) Successive divisionLet us learn these methods by examples.Solved ExampleExample 1 : Find the HCF of 4 and 6. Solution: Method I, [Using set of all factors]Here, F4 = { 1, 2, 4} and F6 = { 1, 2 ,3, 6}Set of common factors = {1,2} Highest common factor = 2.∴ HCF of 4 and 6 is 2.Method 2 [Using prime factors]Here, 4 = 2 × 26 = 2 × 3Common factor = 2∴ HCF of 4 and 6 is 2.2 422 63Acme Mathematics 6 47
Example 2 : Find the HCF of 36 and 42.Solution: In prime factors :36 = 2 × 2 × 3 × 342 = 2 × 3 × 7Common factors = 2 × 3 = 6∴ HCF = 6.Example 3 : Find the maximum number of students to whom 36 mangos and 60 apples can equally be distributed ? How many fruits each get ?Solution: Here, The number of student is the HCF of 36 and 60.36 = 2 × 2 × 3 × 360 = 2 × 2 × 3 × 5Common factor = 2 × 2 × 3 = 12HCF = 12. Hence, There are 12 students.Again there are 36 mangoes and 60 apples. 36– 3612 30 60– 6012 50Here, each student get 3 mangoes and 5 apples.ClassworkFind the HCF of the following numbers:(a) 16 and 20 (b) 24 and 30 (c) 30 and 45 (d) 12 and 15(e) 20 and 25 (f) 21 and 28 (g) 15 and 18 (h) 32 and 36(i) 40 and 70 (j) 46 and 56 (k) 25 and 75 (l) 50 and 100Exercise 2.51. Calculate the HCF of the following numbers using set of all factors.(a) 4,6 (b) 6, 8 (c) 8, 10 (d) 10, 14(e) 6, 8, 4 (f) 10, 12, 8 (g) 6, 9, 3 (h) 10, 15, 5(i) 28, 35 (j) 48, 72 (k) 121, 605 (l) 64, 962 362 183 932 423 2172 602 303 155 2 362 183 93 48 Acme Mathematics 6
2. Find the HCF of the following number using factorization methods.(a) 27, 63 (b) 12, 72 (c) 54, 72(d) 28, 49, 35 (e) 72, 96, 48 (f) 121, 550, 33(g) 36, 60, 88 (h) 49, 70, 77 (i) 96, 106, 192(j) 64, 96, 160 (k) 152, 190, 114 (l) 145, 203, 2613. (a) Find the largest number that divides 85 and 10 without a remainder.(b) Find the largest number that divides 280 and 490 without a reminder.4. (a) An oil merchant has 80 litters of oil in one drum and 88 litters in another drum. What is the capacity of the largest container that can be used to completely empty the oil in the drums ? (b) A box contains 40 oranges and 90 bananas. What is the maximum number of students to whom each fruits can be divided equally ? (c) A large hall of size 24 m by 70 m has to be tiled. What is the length of the largest square tile that can be laid ?5. (a) Find the maximum number of students to whom 12 mangos and 20 apples equally be distributed ? How many fruits do each get ?(b) If 20 books and 180 pens are distributed equally among a group of students, find the biggest number of students in a group.(c) If 45 pencils and 165 copies are distributed equally among a group of girls, find the greatest number of girls in a group.Acme Mathematics 6 49
(b) Lowest Common Multiple (LCM)i. MultiplesMultiplication table gives us the multiple of that number. For example, table of 2 and 3 are:2 → 2 4 6 8 10 12 14 16 18 20 22 243 → 3 6 9 12 15 18 21 24 27 30 33 36Hence, Multiples of 2 are: 2, 4, 6, 8, 10, .............. and so on.Multiples of 3 are: 3 , 6, 9, 12, 15, 18, .......... and so on.I know, every number is multiple of itself.Multiples are never ending.Every number is a multiple of 1.ii. Common multiplesFrom the above multiplication table 6, 12, 18 and 24 are common to both. Thus two or more multiples can have a common multiples. Common multiples of 2 and 3 are 6, 12, 18, ........ and so on.iii. Lowest common multipleMultiples are never ending. It is impossible to find the highest common multiple. We can find the lowest common multiple only.Multiples of 2 = 2, 4, 6 , 8, 10, 12 , 14, 16, 18 , 20, 22, 24 , ......Multiples of 3 = 3, 6 , 9, 12 , 15, 18 , 21, 24 , 27, 30, ......Common multiples = 6, 12, 18, 24, ......Lowest common multiple (LCM) = 6iv. Lowest common multiple using division methodStudy the given example:This method is listing of multiples.50 Acme Mathematics 6