Acme Mathematics 7

Acme Mathematics 7 1Approved by the Government of Nepal, Ministry 7of Education, Science and Technology, Curriculum Development Centre, Sanothimi, Bhaktapur as an additional material.Please scan forE-book

2 Acme Mathematics 7Publisher : 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

Acme Mathematics 7 3I 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

4 Acme Mathematics 7Content:Sets Working Hour: 10 71.1 Revision 1.2 Type of sets (Empty (Null), Singleton (unit), Finite, Infinite, Equal, Equivalent)1.3 Universal set 1.4 Sub-sets Arithmatics Working Hour: 45 242.1 Whole neember 2.2 Integers 2.3 Rational Number2.4 Fraction and Decimal2.5 Ratio and Proportion2.6 Profit and loss2.7 Unitary method Mensuration Working Hour: 20 1283.1 Perimeter of plane figure3.2 Surface area of a cube and a cuboid3.3 Volume of cuboid and cube3.4 Circle

Acme Mathematics 7 5Algebra Working Hour: 30 1634.1 Indices 4.2 Algebraic Expression 4.3 Equation, inequality and graphGeometry Working Hour: 50 2185.1 line and angle 5.2 Triangle and Quadrilateral 5.3 Similar and Congruent Objects5.4 Solids 5.5 Co-ordinates 5.6 Distance between two points5.7 Symmetry and Tessellations 5.8 Transformation5.9 Bearing and Scale DrawingStatistics Working Hour: 10 3006.1 Collection of data6.2 Line graph6.3 Bar Graph

6 Acme Mathematics 7

Acme Mathematics 7 71UNIT Sets1. Look at the given group of pictures carefully and name the sets if possible.(a) (b)(c) (d)2. Write true or false.(a) A class is a set of students. (b) A battalion is a set of army.(c) A geometry box is a set of compass, divider, set-squares, protector, pencil and eraser.(d) Set is a collection of any objects.3. Fill in the blanks. (a) Sets are denoted by ............. .(b) In the set A= {1, 2, 3}, the numbers 1, 2, 3 are called ............ .(c) In the set B = {a, b, c, d}, number of elements = .............. .4. Write the meaning of the given symbols: (a) ∈ (b) ∉ (c) n(A) 5. Give one example of the following sets. (a) Empty set (b) Null set (c) infinite set (d) Unit set 6. List the elements of the following sets. (a) {Days of the week} (b) {Our notes} (c) {Odd number less than 6}7. Write any 3 more sets taking the elements of the following sets. (a) A = {1, 2, 3} (b) X = {Ram, Shyam, Hari, Kalu} (c) B = {all factors of 6} (d) M = {Days of the week}Warm Up Test

8 Acme Mathematics 71.1 RevisionSet is a collection of well defined objects. Sets are denoted by the capital letters like A, B,C..... X, Y, Z and enclosed by curly brackets { }. For example, the set of means of transport.T = {car, truck, bus, aeroplane}The set of vowel sound in English alphabet is denoted by V as, V= {a, e, i, o, u}The order of the members in the set has no meaning. Sets {o, n} or {n, o} carry the same meaning.Consider the set N = {1, 2, 3, 4}.Here, 1, 2, 3 and 4 are the elements of the set N. Thus member of a set is called element of that set. We read N and 1 as; '1 belong to N' i.e. '1' is a member of the set N.'1' is a element of the set N. It is written as, '1'∈N.Thus,2∈N, 3∈N, 4∈N But 5∉N'∈' indicates belongs to.'∉' indicates does not belong to.'5' is not a member (element) of the set N.Sets are described in three ways.(a) Description method:If we describe the character of the members either in word or in sentence, it is called description method.For example: D = {first four even numbers}(b) Listing method:If we write the member inside the curly bracket with comma, it is called listing method.For example: D = {2, 4, 6, 8}(c) Set-builder method:If we write the member as variable (like x, y, z, etc.) and describe the common property of the variable, it is called set-builder method.For example: D = {x:x is first four even numbers}I know ∈ is a Greek letter. ∈ is epsilon.' ' is read as such that

Acme Mathematics 7 9Number of elements on the setsConsider the sets.(a) V = {a, e, i, o, u} (b) N = {1, 2, 3 .......... 50}Here, a, e, i, o, u are the members of the set V. V has 5 members.Similarly set N has 50 members.These can be written as n (V) = 5 and n (N) = 50. 'n' represent the number of elements. n(V) is read as 'The number of elements in the set V'.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 'two'} in listing method.Solution: B = {t, w, o}Classwork1. Write 'True' or 'False' in the blanks.(a) Set is a group of things .................(b) Sets are denoted by English capital letters .........................(c) '∉' is symbol for 'does not belong to' ..............(d) The order of the members in a set has special meaning...............2. If O ={1, 3, 5} and E ={4, 6, 8} are two sets, put the symbols ∈or ∉ in the blanks.(a) 3 ............ O (b) 5 ............. O (c) 8 ........... E(d) 1 ............ $ (e) 6 .............. E (f) 4 ............ O3. Write 'True' or 'False' in the blank.(a) 10 ∈{natural number} .......... (b) 5 ∉{prime number} ..........(c) 2 ∈ {composite number} .......... (d) Red ∈ {traffic light} ..........(e) You ∉ {member of family} ..........

10 Acme Mathematics 7Exercise 1.11. Tick () the well-defined statements.(a) The tallest students. (b) Months of a year. (c) Beautiful flower.(d) Our coins. (e) The best teacher.2. List any four members in the blanks.(a) {Districts of Bagmati Pradesh} .............................(b) {Days of a week} ............................(c) {Prime number less than 10} ...................(d) {Our months} ........................(e) {Our notes}.....................................3. Write 'True' or 'False\".(a) If A = {a, b, c, d}, then set A has 4 members(b) If M = {1,2, 3,.................}, then n(M) = 3(e) If R = {o}, R is not a set.4. Write in description method.(a) {1, 3, 5, 7} (b) {2, 4, 6, 8,10} (c) {cow, dog}(d) {Baishakh, Jestha} (e) {Sunday, Saturday} 5. 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} 6. 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}7. Calculate the cardinality of the following sets.(a) E ={10, 12, 14, 16, 18} (b) A ={ a, b, c, ............ z}(c) B = {2, 4, 6, 8, ........., 20} (d) P ={ Prime number less than 10}(e) A = {Odd numbers between 100 and 108}(f) R = {x:x is number less than 30 and divisible by 2} (g) W = {letters of the words 'our school is a best school'}

Acme Mathematics 7 111.2 Types of sets(a) Empty (Null) setHow many girls are there in the group along side?A set containing no element is called empty set. For example, A = {a 200 years old man}. It is denoted by the symbol φ or { }. A girl in the group of boys is also Null set.(b) Singleton (unit) setA set containing only one element is called singleton set. For example, B = {the highest peak in the world}(c) Finite setA set containing the countable number of elements is called finite set. For example, X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}(d) Infinite setA set containing the uncountable number of elements is called infinite sets. For example, Y = {1,2, 3, 4, .........}S = {stars in the sky}(e) Equal setsTwo sets A and B are said to be equal sets if both have the same elements. For example, if A = {g, o, d} and B ={ d, o, g} then set A and set B are equal sets. X1254367 8910

12 Acme Mathematics 7(f) Equivalent sets Two sets A and B are said to be equivalent sets if the cardinality of sets are equal.For example, if A ={1, 2, 3} and B={a, b, c} then A and B are equivalent sets as n (A) = n(B). It is denoted by A~B.The diagram shows the equivalent sets A and B.A A AA~B A~B A~BB B B123123123abcbcacabClasswork1. Classify finite and infinite sets.(a) X ={1, 2, 3 ...............9} (b) Z ={2, 4, 6, ....20, 22........}(c) A ={points on the line} (d) W = {prime numbers}(e) Y ={set of multiple of 4} (f) B ={all factors of 32}2. Write 'True' or 'False;'.(a) If S = {Sunday, Monday, Saturday}, S is a finite set. (b) If W = {0, 1, 2, 3 ..............}, W is a infinite set. 3. Match the following:Unit set {1, 2, 3, 4, ..........}Finite set {o}Infinite set {a, e, i, o, u}Empty set A and B are not equivalent setn(A) > n(B) { }

Acme Mathematics 7 13Exercise 1.21. Write equal sets for the following given sets A,B,C,D,E,F,W,Z and V.(a) (b) (c)A = {1, 2, 3} B = {r, a , t} C = {first three natural numbers}(d) (e) (f)D = {g, o, d} E = {all factors of 4} F = {o, d, g}(g) (h) (i)W = {1, 3, 5, 7, 11} Z = {0} V = {a, e, i, o, u}2. List the elements and identify the empty set or singleton set in the following.(a) A = The set of women prime minister in Nepal.(b) X = The set of even prime number.(c) B = The set of number neither prime nor composite. (d) C = The set of highest peak in the world.(e) Z = The set of even numbers between 10 and 11.3. Write any one equivalent set for every given sets.(a) D = {Days of a week} (b) E = {0, 2, 4, 6, 8}(c) M = {Multiple of 12 less than 60} (d) A = {January, June, July}(e) X = {0, 1} (f) Z = {z, o}(g) F = {all factors of 15} (h) G = {prime factors of 30}4. If A = {a, b} and B = {c, d}, show the sets on diagram. Are the sets A and B equivalent?5. Study the given sets, A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5}.(a) Are the sets A and B equal?(b) How the sets A and B can be made equal?Project WorkList the name of things around you. Arrange them according to their character. Identify if they are finite or infinite sets.

14 Acme Mathematics 71.3 Universal SetLet us see the following sets: A = {1, 3, 5, 7, 9}and B = {2, 4, 6, 8, 10}. Here, in the sets A and B, the elements are taken from natural numbers up to 10. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Now, from the figure alongside, set A is a subset of U. set B is a subset of U. What can be concluded? A universal set is the set containing all the elements of a particular situation under consideration. Generally it is denoted by the english capital letter U.Solved ExampleExample 1 Write down the possible universal set for the following sets: A ={1, 3, 5} and B = {2, 4, 6} Solution: The universal set for the sets A and B may vary. Some possible universal sets are:U = {Natural number less than 7} or, U = {Whole number less than 7} or, U = {Positive integers} etc.Thus, universal set is not unique. Example 2 If U = {natural number from 10 to 30}, list the elements of the following sets: (a) A = { all the factors of 10 } (b) B = { multiples of 5 } (c) D = { composite numbers } (d) C = { odd composite numbers } Solution: (a) A = {10} (b) B = {10, 15, 20, 25, 30} (c) D = {10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30} (d) C = {15, 21, 25, 27} Here every set A, B, D or C are the sets from the universal set U. Universal set U contains more elements than each set A, B, D or C.Class is a universal set for a student.A B137 952 46 810U

Acme Mathematics 7 15Classwork1. Find the universal set for the following sets and represent them in the Venn diagrams: (a) A = {2, 3, 4, 5, 6}, B = {4, 5, 6, 7, 8} (b) C = {Nepal, India, China}, D = {Nepal, India} (c) E = {a, b, c, d}, F = {a, e, i, o, u} (d) B = { Ram, Shyam, Hari }, G = { Sita, Silu, Sima } (e) N = {Manmohan, Ganeshman, Madan},M = {Bisheswor, Manmohan, Puspalal} 2. State the possible universal set for each of the following sets : (a) A = {a, b, c} (b) B = {2, 4, 6, 8} (c) C = {1, 3, 5, 7} (d) D = {a, e, i, o, u} (e) E = {a, b, c, d, e} (f) F = {Δ, , } 3. State the possible universal set for each of the following sets:(a) A = {Chitwan, Makawannpur, Bara}(b) X = {Narayani, Trisuli, Kaligandiki} (c) Y = {Sagarmatha, Kanchanjangha, Machhapuchhre} (d) Z = {Sunday, Thursday, Saturday} Exercise 1.31. Write down a possible universal set for each of the following sets: (a) A = {compass, ruler, set square, protractor} (b) C = {rat, cat, bat} (c) B = {point, line, ray, line segment} (d) D = {tiger, rhino, deer} (e) X = {shirt, paint, coat, sweater} (f) E = {pen, pencil, chalk} 2. Write down the single universal set for each of the following sets from No,(a) to (f): (a) {2} (b) {2, 4, 6} (c) {8, 10, 12} (d) {3, 5, 7, 9, 11} (e) {3, 6, 9, 12} (f) {6, 7, 8, 9} 3. Which of the following sets is the universal set of {1, 3, 7, 8}? (a) {2, 3, 5} (b) {0, 1, 2, 7} (c) {0,1, 2, 3, 4, 5, 6, 7, 8}4. Suggest any two universal sets for each of the following sets: (a) X = {all factors of 12} (b) Y = {Earth, Mars, Jupiter} (c) Z = {cycle, bus, car, train} (d) A = {boys of class 7}

16 Acme Mathematics 7(e) B = {orange, apple, banana, grapes} 5. If U = {10, 11, 12, ......... 20} then, list the following sets: (a) A = {x : x is even numbers} (b) X = {x : x is odd numbers} (c) Z = {x : x is square numbers} (d) Y = {x : x is prime numbers} (e) B = {x : x is composite numbers} (f) D = {x : x is multiples of 25} 6. If U = {x : x ∈ N, x < 10} be a universal set , A = {y : y is even number} and B = { z : z is prime number}, (a) List the elements of U. (b) List the elements of A. (c) List the elements of B. (d) Are the sets A and B equal? (e) Are the sets A and B equivalent? 7. If U = {x : x ∈ N, x < 12} be a universal set , A = {y : y is odd number} and B = { z : z is even number}, (a) List the elements of U. (b) List the elements of A. (c) List the elements of B. (d) Are the sets A and B equal? (e) Are the sets A and B equivalent? 8. 10. If U = {x : x ∈ N, x < 15} be a universal set , A = {y : y is even number} and B = { z : z is prime number},(a) List the elements of U. (b) List the elements of A. (c) List the elements of B. (d) Are the sets A and B are equal? (e) Are the sets A and B equivalent? 9. If U = {a, b, c, ....., h, i} be a universal set, A = {vowel alphabets} and B = { consonant alphabets}, (a) List all the elements of U. (b) List the elements of A. (c) List the elements of B. (d) Are the sets A and B equal? (e) Are the sets A and B equivalent?10. Study the following condition and write a universal set U.(a) There are 4 prime number(b) There is 1 even number also.(c) It is less than 9.

Acme Mathematics 7 171.4 Subsets(a) Introduction to subsets Look at the following set of friends. F is a set of two friends Ram and ChandraSo, F = {Ram, Chandra} We can use the member of the set F to make the following sets: (i) A = {Ram} (ii) B = {Chandra} (iii) C = {Ram, Chandra} (iv) N = { } or φHere, sets A, B and C are contained in the set F. These sets are called the sub-sets of the given set F. Set A is said to be a subset of a set B if every member of set A is also a member of set B. For example, If A = { 2, 3, 5 } and B = { 2, 3} are two sets then each member of set B is also a member of set A. Hence, set B is a subset of set A. Symbolically it is written as B ⊂ A where we read '⊂' as \" is a subset of \"or \" contained in\". Study the following cases of subsets. Case I: Proper subset Consider the sets, A = {2, 3, 5} and B = {2, 3}. Here, each elements of set B is also an element of set A. Since number of elements in set B is less than set A, in such a case we say that set B is the proper subset of set A and we write it as B ⊆ A.Case II: Improper subset Consider the sets, A = {2, 3, 5} and B = {5, 3, 2}. Here, each element of set B is also an element of set A and both has equal number of elements.Since, n(A) = n(B), in such a case we say that B is an improper subset of set A and we write it as B ⊆ A. Thus, a set is an improper subset of itself. A B2 23 35AB2 352 35B A

18 Acme Mathematics 7Case III: Consider the sets, A = {multiple of 5 less than 4} and B = {1, 2, 3, 4}. Here, set A has no element, which is also an element of set B and n(B) = 4 and n(A) = 0. Since n(A) < n(B), in such case, we say that set A is subset of set B and we write it as A ⊂ B or φ ⊂ B. Thus, every empty set is a subset of given set. Note: If set A is subset of set B then the set B is called the super set of the set A. It is denoted by B ⊃ A and read as 'B is super set of the set A'. Solved ExampleExample 1 Let N = {1, 2, 3, 4, 5} is a given set. List the following sets: (a) O = Set of odd numbers (b) E = Set of even numbers (c) W = Set of whole numbers (d) M2 = Set of multiples of 2 (e) M12 = Set of multiples of 12 (f) F5 = Set of factors of 5 Solution: (a) O = {1, 3, 5} (b) E = {2, 4} (c) W = {1, 2, 3, 4, 5} (d) M2 = {2, 4}(e) M12 = { } or φ (f) F5 = {1, 5}Example 2 Study the following dairy items milk, paneer, butter Choose at least 2 items to prepare breakfast you like most.Solution: Here,set of dairy items, D ={milk, panner, butter}break fast items is subset of DSo, possible choice are {milk, panner}, {milk, ghee} and {paneer, ghee}.Classwork1. If A = {d, e, f}, and C = {a, b, c, d} then find the following : (a) Write a subset of C. (b) Write two subset of A.(c) Write 3 subset of C. (d) Write 4 subset of C.(e) Write all subset of A.1 234BA

Acme Mathematics 7 192. Here is the menu of \"Malewa\" restaurant at Bharatpur in Chitwan: (a) Buff momo (b) Pizza (c) Veg momo (d) Finger chips (e) Chow mein (f) Green Salad (g) Chicken soup (h) Bhatmas Sandeko (i) Mushroom Soup(j) Kaju fry (k) Chicken chilly (l) Meat Prepare your 'Khaja Menu' for 10 days taking only two items at a time.Exercise 1.41. The set of students of a class is given below: U = {Ram, Narayan, Yadav}. List the following subsets of U. (a) whose name starts with R (b) whose name starts with N (c) whose name starts with Y (d) whose name starts with R and N (e) whose name starts with R or Y2. Let A = {1, 3, 5, 7} be the given set list the following subsets of A: (a) Q = Set of odd numbers (b) P = Set of prime numbers (c) C = Set of composite numbers (d) F3 = Set of factors of 3 (e) M2 = Set of multiples of 2 (f) M7 = Set of multiples of 73. Write all the proper subsets of the set W = {p, q, r}. 4. Write all the subsets of the set P = {19, 20, 21}. 5. Write the number of the subsets of the following sets: (a) {1} (b) {a, b} (c) {x, y, z} 6. Look at the following items of breakfast list of a hostel: hot-lemon, milk, coffee, milk tea, black tea, jam, butter, bread, toast, rotiPrepare a list for a week taking suitable two items at a time. 7. Consider a set of fruits F = {mango, orange, apple}. Make a list of fruits that contains at least two fruits at once.

20 Acme Mathematics 7Project Work1. Objective : To make sub sets.2. Materials required :“ A-4 Size Paper“ Pen3. Activity : “ List the things in your kitchen. Classify them according to same character. Name the sets and three sub sets from each sets. Present it to your class room.For example : Kitchen items (U) : .........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................Sub set A : ..........................................................................................................................................................................................................................................Sub set B : ..........................................................................................................................................................................................................................................Sub set C : ..........................................................................................................................................................................................................................................

Acme Mathematics 7 211. (a) What is finite set?(b) If P = {1, 2, 3}write the subsets of set P. (c) Is the set Q = {1, 2, 3, 4}, the proper subset of P? Give reason.2. If A = {factors of 15} and B = {factors of 18}, then(a) Write the listing method of A and B.(b) Find the cardinal number of A and B. (c) Write the definition of universal set.3. If P = {prime number less than 10} and Q = {a, b, c, d} then, (a) Write set P by listing method.(b) Set P and Q are equal sets: Why? (c) Write improper subset of set P.4. (a) If A = {1, 2, 3, 4, 5}, B = {6, 7, 8, 9} and C = {1, 3, 5} then find the universal set of D having 10 elements. (b) Write all the possible subsets of set M = {a, b}.(c) Is the set M, the proper subset of A? Give reason.5. The following three sets of students of class 7 based on their roll numbers are formed. A = {roll numbers of students less than 5}, B = {roll number of student between 7 and 8}and C = {prime roll numbers of students between 10 and 20}.(a) Which of the above sets is the empty set?(b) State with reason whether the sets A and C are equal or equivalent sets.(c) Write any one subset of set-A consisting of only two elements.6. Three sets P = (x:x is a prime number less than 10), Q = {2, 3, 5, 7}and R = {natural number between 5 and 6} are given,(a) Rewrite the statement \"9 does not belong to set Q. Using set notation.(b) List the elements of set P.(c) Is set R a null set? Give reason.7. If set A = {a, b, c} and B = {b, c},(a) Write the number of possible subset of B.(b) Write all the possible subsets of A.(c) Write the relation of set A and B in set notation.Mixed Exercise

22 Acme Mathematics 78. Set P = {Prime numbers less than 10}, Q = {2, 3, 5, 7} and R = {Natural numbers between 5 and 10} are given(a) Define subset with an example.(b) Are set P and set Q equal set? Justify.(c) Find the number of proper subset formed by set R.9. A given set P = {Prime Number less than 20}.(a) List the elements of this set.(b) What is the cardinal number of the set?(c) Is the set 'p' finite or infinite set? Give reason.10. If A = {x : is odd number less than 6}(a) Write set A by listing method.(b) Write a proper and improper subsets of set A.(c) Write a set which is equivalent to set A.11. N = {Natural numbers less than 10}, E = {Even number less than 10}, P = {Prime numbers less than 7}, F = {Factors of 9} are given.(a) Write any two subset of the set N.(b) Which of the four sets are equivalent sets?(c) Is the set E being the proper subset of set N? Write with reason.12. Set A = {First five odd numbers}, B = {vowel letters} and R = {2, 3, 5, 7, 11, 13, 17, 19} are given.(a) List the elements of set A and set B.(b) Are set A and set B equivalent set? Justify.(c) Is set A ⊆ B. Why?13. If A = {The set of prime numbers less than 10}, B = {The set of odd numbers less than 10}.(a) Define cardinality of any set.(b) List the elements of both sets A and set B.(c) Represent the relation of A and B in Venn-diagram.14. (a) If A = {1, 2, 3} and B = {4, 5, 6} then A and B are equal set or not?(b) Write the universal set for set A and B.(c) A set having no element is known as ........ set.15. (a) If P = (x : x is even prime number}, then write the type of set P. (b) Define finite set with an example.

Acme Mathematics 7 23EvaluationTime: 30 minutes Full Marks: 121. If M = {a, e, i, o, u} and N = {a, b, c, d, e} are two sets then answer the following questions. (a) Which types of sets M and N are equal or equivalent ? Give reason. [1](b) Write the universal set of M and N. [1](c) Write the improper subset of set M. [1]2. P = {prime number less than} and Q = {a, b, c, d} are given. (a) Write set P by listing method. [1](b) Set P and Q are equal sets, why ? [1](c) Write improper subset of set P. [1]3. Sets P = {a, b, c, d}, Q = {a, b} and R = {e, f, g, h} are given. (a) Are the set P and set R equivalent or equal set, why ? [1](b) Making the universal set of set P, Q and R. [1](c) Which set is the proper subset of P ? [1]4. Set A = {factor of 6} and B = {2, 4, 6, 8} are given. (a) Write set A by listing method. [1](b) Are the set A and set B finite of infinite write ? [1](c) Define null set. [1]

24 Acme Mathematics 72UNIT Arithmetic1. Fill in the blanks: (a) 1 = 12, 100 = 102, 10000 = 1002 (b) 4 = .................., 400 = .................., 40000 = ...................... (c) 9 = .................., 900 = .................., 90000 = ...................... (d) 16 = .............., 1600 = .............., 160000 = ................... 2. Complete the pattern. (a) 1 + 2 + 1 = 22 (b) 1 = 121 + 2 + 3 + 2 + 1 = 32 1 + 3 = 221 + 2 + 3 + 4 + 3 + 2 + 1 = 42 1 + 3 + 5 = 32........................ = 52 ........................ = 42........................ = 62 ........................ = 52........................ = 72 ........................ = 62(c) 12 = 1 ⇒ 1 = 12 112 = 121 ⇒ 1 + 2 + 1 = 22 1112 =12321 ⇒ 1 + 2 + 3 + 2 + 1= 32................ = 1234321 ⇒ ................ = 42 ................ = ................ ⇒ ................ = 523. Find the square of: (a) 5 (b) 7 (c) 17 (d) 22 4. Find the square root of: (a) 100 (b) 625 (c) 169 (d) 225 5. Fill in the blank: (a) 13 = .................., 23 = .................., 33 = ..................... (b) 103 = .................., 203 = .................., 303 = ..................... (c) 1 3 = .................., 8 3 = .................., 27 3 = .................. 6. Complete the pattern. (a) 13 = 12 = 1 13 + 23 = 32 = 9 13 + 23 + 33 = 62 = 36 ....................... = ....................... = ....................... ....................... = ....................... = ....................... 7. Find the cube of: (a) 5 (b) 7 (c) 2 (d) 11 8. Find the cube root of: (a) 64 (b) 125 (c) 216Warm Up Test

Acme Mathematics 7 25A Square and square root1. SquareLook at the following table and dot pattern:Numbers Square of Numbers1 – 1 12 – 2 43 – 3 94 – 4 165 – 5 256 – 6 367 – 7 498 – 8 649 – 9 8110 – 10 1001 4 9 16 25If we can express a number in a dot pattern in the form of a square then the number is called a perfect square number. For example: 9 is a perfect square number because it can be expressed as the pattern We also have, 9 = 3 × 3 = 32.Thus, if we can express the number into the product of the two identical numbers exactly, then the number is called a square number.e.g. 49 is a square number because 49 = 7 × 7 = 72.2.1 Whole Number

26 Acme Mathematics 7Solved ExampleExample 1 Find the square of : (a) 12 (b) 15 (c) 11 (d) 17 Solution: (a) Here, given number = 12 ∴ Square of 12 = 122 = 12 × 12 = 144 (b) Here, given number = 15∴ Square of 15 = 152 = 15 × 15 = 225 (c) Here, given number = 11∴ Square of 11 = 112 = 11 × 11 = 121 (d) Here, given number = 17 ∴ Square of 17 = 172 = 17 × 17 = 289 Example 2 Which of the following numbers are perfect square? (a) 100 (b) 175 (c) 225 Solution: (a) Here, 100 = 2 × 2 × 5 × 5 = 22 × 52= (2 × 5)2= 10 × 10 = 102∴ 100 is a perfect square number. (b) Here, 175 = 7 × 5 × 5 = 7 × 527 has no pair∴ 175 is not a perfect square number. (c) Here, 225 = 3 × 3 × 5 × 5 = 32 × 52= (3 × 5)2 = 152∴ 225 is a perfect square number. Example 3 Find the least number by which 1575 is multiplied so that the product is a perfect square. Solution: Here, 1575 = 3 × 3 × 5 × 5 × 7 = 32 × 52 × 7 Here 3 and 5 have the pairs but 7 has no pair. So, if we multiply 1575 by 7, then the 7 has also pair. Therefore, the required least number is 7.3 15753 5255 1755 357

Acme Mathematics 7 27Classwork1. Write true of false for the following statements:(a) A square number always has an even number of digits.(b) Square number are always positive.(c) There is no square number between 100 and 125.(d) Square of an even number is another even number.(e) Square of a prime number is also a prime number.(f) Product of two square numbers is also a square number.(g) Sum of two square numbers is also a square number.(h) Difference of two square numbers is also a square number.2. Simplify: (a) 132 (b) 182 (c) 212 (d) 1002 (e) 1252 (f) 100023. Verify that :(a) 32 + 42 = 52 (b) 122 + 52 = 132(c) 102 – 82 = 62 (d) 152 – 92 = 122Exercise 2.11. Calculate the square of: (a) 14 (b) 18 (c) 27 (d) 32(e) 35 (f) 39 (g) 43 (h) 49 (i) 58 (j) 65 (k) 64 (l) 70(m) 80 (n) 92 (o) 111 (p) 257 (q) 350 (r) 512 2. Express in the square form: (a) 225 (b) 289 (c) 484 (d) 1225(e) 1600 (f) 3249 (g) 3600 (h) 5776(i) 6400 (j) 10000 (k) 12544 (l) 22500 3. Determine which of the following numbers are perfect square number. (a) 625 (b) 1250 (c) 140 (d) 729 (e) 784 (f) 1000 (g) 1440 (h) 16900(i) 22500 (j) 12321 (k) 46225 (l) 56782

28 Acme Mathematics 74. By which the smallest number should the following numbers be multiplied in order to make the product perfect square numbers? (a) 12 (b) 45 (c) 63 (d) 108(e) 288 (f) 605 (g) 216 (h) 1260 (i) 2592 (j) 2700 (k) 1500 (l) 3000 5. By which smallest number should the following numbers be divided so as to get the perfect square number? (a) 75 (b) 180 (c) 242 (d) 9075(e) 3564 (f) 768 (g) 2268 (h) 1875(i) 8640 (j) 4500 (k) 2250 (l) 3861 6. (a) If the side of a square room is 17 m, find the area of the room. (b) If there are as many students as the rupees that each get, find how much money is needed to distribute among 49 students. 7. (a) What is the perfect square number which has 19 as one of the two equal factors?(b) What is the perfect square number which has 14 as one of the number ? 8. (a) What is the smallest number by which 8575 must be divided in order to become a perfect square?(b) What is the smallest number by which 3645 must be divided in order to become a perfect square? (c) What is the smallest number by which 6300 must be divided in order to become a perfect square?9. (a) What is the smallest number by which 1152 must be multiplied in order to become a perfect square ?(b) What is the smallest number by which 4375 must be multiplied in order to become a perfect square ?(c) What is the smallest number by which 3072 must be multiplied in order to become a perfect square ?

Acme Mathematics 7 29Project Work1. Objective : To find the properties of square numbers.2. Materials required :A-4 Size Paper Colour Pencil Scale3. Activity : “ Draw the table. Write the numbers from 1 to 40 and their square numbers.“ Circle the unit digit of the numbers.“ Circle the unit digit of the square numbers.For example : Number Square Number Square Number Square Number Square1 11 21 312 12 22 323 13 23 334 14 24 345 15 25 356 16 26 367 17 27 378 18 28 389 19 29 3910 20 30 40Now answer the following question.a. Does square number end with 2, 3, 7 or 8 ?b. Does square number end with 0, 1, 4, 5, 6, or 9?c. If the number ends with 5, its square also ends with 5, is it?Write true or false.a. If a number ends with 0, then it's square also ends with 0.b. Square of an odd number is also odd number.c. Square of an even number is also even number. d. Product of any two square number is always square number.

30 Acme Mathematics 72. Square rootStudy the following table:(i) Square number 100 81 64 49 36 25 16 9 4 1Square root of the number ± 10 ± 9 ± 8 ± 7 ± 6 ± 5 ± 4 ± 3 ± 2 ± 1(ii) Look at the following dot pattern:From the above table we have,100 = 10 × 10 = 102 or, – 10 × – 10 = (– 10)281 = 9 × 9 = 92 or, – 9 × – 9 = (– 9)264 = 8 × 8 = 82 or, – 8 × – 8 = (– 8)2Since, 100 = 102 or, (– 10)2Square root of 100 = (10)2 or, (– 10)2= 10 or, – 10∴ 100 = 10 or – 10 = ± 10Similarly, 81 = ± 9,64 = ± 8 and so on.Square root of a number is a number which when multiplied by itself, gives the given number. If x is a number, then its square root is denoted by x, where is radical sign and it indicates the square root of the given number.Note: 2 is radical sign for square root but in general we use only. The positive square root is called principal square root.Methods to find the square rootWe will now learn three methods to find the square root of a given number. These methods are : (a) Repeated subtraction methodWe will use this properties to find the square root of a given perfect square number. The rule is as follows.From the given number we subtract the odd numbers 1, 3, 5, 7, 9, ... one by one. If the given number is a perfect square, we will get zero after some subtractions. When we get

Acme Mathematics 7 31a zero, we count the number of times subtraction is done and write it as the square root of the given number. For example : consider the number 25.subtract first odd number, 1 = 25 – 1 = 24from the result subtract second odd number, 3 = 24 – 3 = 21from the result subtract third odd number, 5 = 21 – 5 = 16from the result subtract fourth odd number, 7 = 16 – 7 = 9from the result subtract fifth odd number, 9 = 9 – 9 = 0Now, subtraction is done 5 times to obtain 0. Therefore square root of 25 is 5.(b) Prime Factorization Method: Steps:(i) find the prime factors of the number (ii) put in squared form (iii) take one number from each squared form and multiply if needed. The example given below gives the idea about the method.Example 1 Find the square root of 64. Solution: Here, 64= 2 × 2 × 2 × 2 × 2 × 2= 22 × 22 × 22= 2 × 2 × 2 = 8∴ The square root of 64 is 8.(c) Division Method: Steps:(i) Make pairs of the numbers from the right and put a bar. (ii) Choose the first pair (or a single) number and guess its nearest square root number. (iii) Divide the first pair by the square root of the nearest square number. (iv) Follow the rule of division and repeat the process.The example given below gives the idea about the method. 2 642 322 162 82 42 21

32 Acme Mathematics 7Example 1 Find the square root of 144. Solution: Hence, square root of 144 is 12.Example 2 Simplify: 102 – 82Solution: Here, 102 – 82 = 10 × 10 – 8 × 8= 100 – 64= 36 = 6 × 6 = 6∴ 102 – 82 = 6Example 3 Find the product and simplify : 36× 32 × 2Solution: Here, 36× 32 × 2= 36 × 32 × 2= 2 × 2 × 3 × 3 × 2 × 2 ×2 × 2 × 2 × 2= 22 × 32 × 22 × 22 × 22= 2 × 3 × 2 × 2 × 2 = 48∴ 36 × 32 × 2 = 48Example 4 If 900 students are arranged in equal number of rows and columns, find the number of students in each row. Solution: Here, number of rows and columns are equal so the number of students in each row is the square root of 900. Now, The square root of 900 = 900= 2 × 2 × 3 × 3 × 5 × 5= 22 × 32 × 52= 2 × 3 × 5 = 30∴ The number of students in each row is 30.1+ 11 44– 11222 44+ 2 – 4424 02 362 183 932 322 162 82 42

Acme Mathematics 7 33Example 5 What is the smallest number that must be subtracted from 8575 so that the difference is a perfect square?Solution:Hence , 111 must be subtracted from 8575. Difference = 8575 – 111 = 8464. 8464 is a perfect square number.Classwork1. Write true of false for the following statements:(a) There are 4 digits in the square root of 1550025.(b) There are 2 digits in the square root of 10201.(c) If a number is negative then its square is also negative.(d) 0.4 = 0.2 2. Fill in the blanks:(a) Square of an odd number is .......... number.(b) 0.25 = ............ (c) 10000 = ............ (d) (0.07)2 = ............3. Simplify: (a) 11 + 13 + 15 + 17 + 8 (b) 52 – 42 (c) 12 + 22 + 32(d) 132 – 122 (e) 32 + 42 (f) 252 – 72Exercise 2.21. Using repeated subtraction method find the square root of the following numbers.(a) 16 (b) 49 (c) 81 (d) 64 (e) 100 (f) 1442. Find the square root of: (a) 121 (b) 324 (c) 1225 (d) 1764 (e) 2704 (f) 2916(g) 3025 (h) 3600 (i) 4096 (j) 4761 (k) 4900 (l) 5184 3. Find, using prime factors, the square root of: (a) 4 × 16 (b) 25 × 64 (c) 49 × 100 (d) 144 × 36 (e) 361 × 81 (f) 36 × 49 × 1009+ 985 75– 8192182 475+ 2 – 364184 111

34 Acme Mathematics 7(g) 25 × 144 × 121 (h) 169 × 256 × 4 (i) 225 × 49 × 36(j) 100 × 625 × 25 (k) 4096 × 4 × 1 (l) 6561 × 81 × 254. Simplify: (a) 484 (b) 529 (c) 961 (d) 1849 (e) 1089(f) 18 (g) 1008 (h) 1536 (i) 476 (j) 5408(k) 14400 (l) 156255. Simplify: (a) 18 × 2 (b) 5 × 125 (c) 3 14 × 28(d) 7 × 2 21 (e) 63 × 27 × 35 (f) 20 × 72 × 6(g) 15 × 24 × 10 (h) 50 × 98 × 32 (i) 72 × 128 × 186. Simplify: (a)3 × 202 × 5 (b)63273 (c) 45 ÷ 45(d)8751125 (e)4872 (f)562947. Find the square root of:(a) 100.4 (b) 0.640.09 (c) 2.25 (d) 225100(e)9.006.25 (f) 0.0081 (g) 0.0625 (h) 9256(i) 1144 (j) 0.00410 (k) 0.000001 (l) 0.091008. Find the length of the following square whose area is given in the figure: (a)625 cm2(b)2601 cm2(c)3249 cm29. (a) Rs. 4624 was distributed among as many rupees as students there were. How many students were there? (b) Rs. 20736 was distributed as many rupees as there were students. How many students were there?

Acme Mathematics 7 3510. (a) 1369 students are arranged in a form of a square. Find the number of students in each row and number of rows? (b) A military general arranges 5650 soldiers in the form of a square so that 25 soldiers are left after the arrangement. Find the number of rows and the number of soldiers in each row. 11. (a) What number, when multiplied by it, gives the product 3364? (b) What number multiplied by itself gives the product 9216 ?(c) What number multiplied by itself gives the product 5625 ?12. (a) By which least number 4050 be multiplied to make it a perfect square? (b) By which least number 6912 be multiplied to make it a perfect square? 13. (a) By which least number 12288 be divided to make it a perfect square? (b) By which least number 25920 be divided to make it a perfect square?14. (a) What is the smallest number that must be subtracted from 8575 so that the difference is a perfect square? (b) What is the smallest number that must be subtracted from 3645 so that the difference is a perfect square? 15. What smallest number must be added to 300 so that the sum is a perfect square? 16. Find the largest 5 digit number which is a perfect square.17. The area of a square field is 48400 m2. Find the perimeter of the field.18. For the annual function, arrange 45 chairs in each row. If number of rows is equal to the number of chairs in each row, find the total number of chairs.19. A society started a campion 'Plant the Plants'. They bought 750 'Ashoka' plants and wanted to arrange maximum possible number of 'Ashoka' plants in the ground such that the number of row is equal to the number of 'Ashoka' in each row. Find the number of rows if 21 'Ashoka' is left out after the arrangement.20. The length of square field is 40 m. Calculate its area.21. The square hall is 2601 m2 in area. How long is it?

36 Acme Mathematics 7B Cubes and Cube roots1. CubeConsider the following: 23 = 2 × 2 × 2 = 8 33 = 3 × 3 × 3 = 27 43 = 4 × 4 × 4 = 64 The above statements can also be expressed by saying that; the cube of 2 is 8, the cube of 3 is 27 and the cube of 4 is 64.Thus, the cube of any number is that number which is obtained by raising to power 3 to the number. The numbers 8, 27 and 64 are called perfect cube numbers.Solved ExampleExample 1 Find the cube of 7. Solution: The cube of 7 = 73= 7 × 7 × 7 = 343 ∴ The cube of 7 is 343.Example 2 Is 128 a perfect cube number? Solution: Now, 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 23 × 23 × 2= 43 × 2 While regrouping the prime factors of 128 keeping the identical three factors together, we found 2 left behind. Therefore 128 is not a perfect cube number.2. Cube rootWe have seen that 8, 27, 64 and 343 are perfect cube numbers. i.e. 8 = 23, 27 = 33, 64 = 43, 343 = 73We say that, 2 is the cube root of 8, 3 is the cube root of 27, 4 is the cube root of 64 & 7 is the cube root of 343. The cube root of a number is indicated by the symbol 3 .For example, The cube root of 8 is written as 8 3 , Thus 8 3 = 2. Similarly, 27 3 = 3, 64 3 = 4 and 343 3= 7A natural number 'x' is the cube root of a number 'y' if y = x3 or x = y3 . 3 is called a radical, y is called the radicand and 3 is called the index of the radical.2 cm2 cm2 cm2 1282 642 322 162 82 42

Acme Mathematics 7 37Solved ExampleExample 1 Find the cube root of 125. Solution: The cube root of 125 = 125 3 = 5 × 5 × 5 3 = 5Hence, the cube root of 125 is 5. Example 2 Find the smallest number which, when multiplied to 675, will make the product a perfect cube number. Find the cube root of the product. Solution: Now, 675 = 3 × 3 × 3 × 5 × 5 = 33 × 52Here, 5 has no power cube so 675 must be multiplied by 5 to make it a perfect cube number. Again, we have 675 × 5 3 = 33 × 52 × 5 3= 33 × 5 3 3= 3 × 5 = 15∴ The cube root of the product 675 × 5 is 15.Classwork1. Write true of false for the following statements:(a) The cube of an odd number can be an even number.(b) There is no perfect cube whose unit digit is two.(c) There are 10 perfect cubes between 0 and 1000.(d) The product of two perfect cubes is a perfect cube.(e) The cube root of 144 is 12.(f) The cube root of 2197 has two digits.2. Express in the form of cube: (a) 64 (b) 125 (c) 343 (d) 216 (e) 7293. Fill in the blanks:13 = 12 = 1213 + 23 = (1 + 2)2 = 3213 + 23 + 33 = (1 + 2 + 3)2 = 62............................................................................................................................................................................................................................................................................5 1255 2553 6753 2253 755 255

38 Acme Mathematics 7Exercise 2.31. Find the cube root of: (a) 125 (b) 216 (c) 512 (d) 1000(e) 1331 (f) 2197 (g) 3375 (h) 1728(i) 4096 (j) 8000 (k) 9261 (l) 50653(m) 79507 (n) 117649 (o) 175616 (p) 140608 2. Find the cube root of:(a) 216343 (b) 5121000 (c) 7291331 (d) 17282744 (e) 17282197(f) 33756859 (g) 926117576 (h) 1064842875 (i)2979135937 (j) 409642875(k) 0.216 (l) 0.512 (m) 1.331 (n) 1.728 (o) 3.375(p) 15.625 (q) 42.875 (r) 10.648 (s) 0.000008 (t) 0.000001(Hint: 1.331= 13311000 )3. Which of the following numbers are perfect cubes? (a) 3072 (b) 32768 (c) 3888 (d) 46656(e) 10985 (f) 296352 (g) 132651 (h) 314928 4. Find the length of each side of the cube whose volumes are as follows.(a) (b) (c) (d)V = 4096 cm3 V = 42875 cm3 V = 17282744 cm3 V = 0.027 cm35. Find the smallest number which, when multiplied by 256, will make the product a perfect cube. Find the cube root of the product also.6. Find the smallest number, by which when 128 is divided, will make the quotient a perfect cube. Find the cube root of the quotient also.7. A cubical tank has volume 4913 m3. How high is the tank?8. The length, breadth and height of a water tank are equal. If the length of the tank is 10 m. How much water does it can hold? [Hints : 1 m3 = 1000 l]

Acme Mathematics 7 39Project Work1. Objective : To find the properties of cube numbers.2. Materials required :“ A-4 Size Paper“ Colour“ Pencil“ Scale3. Activity : “ Draw the table. Write the numbers from 1 to 30 and their cube numbers.“ Circle the unit digit of the numbers.“ Circle the unit digit of the cube numbers.For example : Number Cube number Number Cube number Number Cube number1 11 212 12 223 13 234 14 245 15 256 16 267 17 278 18 289 19 2910 20 30

40 Acme Mathematics 74. Study the above table and do the following:Write true or false.(a) If a number ends with 1, 4, 5, 6, 9 or 0 then its cube also ends with 1, 4, 5, 6, 9 or 0 respectively.(b) If a number ends with number 2 then its cube ends with 8.(c) If a number ends with number 8 then its cube ends with 2.(d) If a number ends with number 3 then its cube ends with 7.(e) If a number ends with number 7 then its cube ends with 3.(f) The cube of an odd number is an odd number.(g) The cube of an even number is an even number.(h) The product of cubes is always a cube number.Complete the pattern(a) 13 = 1= 1(b) 23 = 8 = 3 + 5(c) 33 = 27 = 7 + 9 + 11(d) 43 = 64 = 13 + 15 + 17 + 19(e) 53 = 125 = 21 + 23 + 25 + 27 + 29(f) ......................................................................................................................(g) ......................................................................................................................(h) ......................................................................................................................(i) ......................................................................................................................(j) ......................................................................................................................(k) ......................................................................................................................(l) ......................................................................................................................n3 = 1 + 3 + 5 + 7 + 9 + ............... + n consecutive odd number.

Acme Mathematics 7 41C Highest Common Factor (HCF) and Lowest Common Multiple (LCM)Warm Up Test1. Fill in the blanks: (a) 232 is divisible by .................... (b) 183 is divisible by .................... (c) 784 is divisible by ............... and ................ 2. Write the first three multiples of: (a) 2 (b) 3 (c) 5 3. Write all the factors of the following numbers. (a) 18 (b) 24 (c) 20 (d) 28 4. Write the prime factors of the following numbers. (a) 20 (b) 48 (c) 36 (d) 60 5. Fill in the blanks. (a) The HCF of 2 and 4 is ....................... (b) The HCF of 3 and 6 is ........................ (c) The HCF of 10 and 20 is ................... (d) The LCM of 2 and 3 is ....................... (e) The LCM of 7 and 8 is ...................... (f) The LCM of 6 and 11 is ..................... 6. Find the HCF of the given numbers. (a) 4 and 6 (b) 8 and 12 (c) 6 and 18 (d) 9 and 15 (e) 12 and 36 (f) 27 and 36 7. Find the LCM of the following numbers. (a) 4 and 6 (b) 6 and 12 (c) 9 and 18 (d) 10 and 30 (e) 11 and 22 (f) 8 and 10

42 Acme Mathematics 71. Highest Common Factor (HCF)In class six, we learnt to find out the HCF of numbers. Now, in this class we discuss different methods to find the HCF. Consider the numbers 15 and 18. All the factors of 15, F15 = {1, 3, 5, 15} All the factors of 18, F18 = {1, 2, 3, 6, 9, 18} Common factors of 15 and 18 = {1, 3} Greatest common factor = 3 ∴ HCF of 15 and 18 is 3.The greatest common factor among the factors of the given numbers is the HCF. The HCF exactly divides the given numbers.The following method helps us to find the HCF of the numbers. (a) Factorization method (b) Definition method (c) Division method The examples given below give the idea about the methods.Solved ExampleExample 1 Find the HCF of 18, 36 and 72 using, (a) Factorization method (b) Definition method(c) Division method Solution: (a) Factorization method Here, 18 = 2 × 3 × 3 36 = 2 × 2 × 3 × 372 = 2 × 2 × 2 × 3 × 3 Common factors = 2 × 3 × 3 = 18 ∴ HCF of 18, 36 and 72 is 18 (b) Definition MethodHere, All the factors of 18, F18 = {1, 2, 3, 6, 9,18} All the factors of 36, F36 = {1, 2, 3, 4, 6, 9, 12, 18, 36} All the factors of 72, F72 = {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36,72} Common factors of F18, F36 and F72 = {1, 2, 3, 6, 9, 18} Here, highest common factor = 18 ∴ HCF of 18, 36 and 72 is 18. A B5 6U3

Acme Mathematics 7 43(c) Division MethodHere, given numbers are 18, 36, 72. Dividing 36 by 18 we have, HCF of 18 and 36 is 18, again dividing 72 by the HCF 18 We have, 18) 72 (4 – 72 0 0∴ HCF = 18 (Since 18 exactly divides 18, 36 and 72.) Example 2 Find the greatest number which exactly divides 294 and 714. Solution: The greatest number is the HCF of 294 and 714. Given numbers are 294 and 714. Using division method, we have, 294) 714 (2 – 588 126) 294 (2 – 25242) 126 (3 – 126 000HCF = 42 ∴ The required number is 42. Example 3 Among how many students can 275 mangoes and 385 oranges be equally distributed? Find the number of mangoes and oranges received by each student. Solution: The number of students is the HCF of 275 and 385, so using division method we have, 275) 385 (1 – 275 110) 275 (2 – 22055) 110 (2 – 110 000Here HCF is 55. Hence the number of students = 55 18) 36 (2 – 36 0

44 Acme Mathematics 7Now, Number of mangoes received by each student = 27555 = 5 Number of oranges received by each student = 38555 = 7 ∴ There are 55 students and each student gets 5 mangoes and 7 oranges.Classwork1. Match the following:all factors of 30 2, 3, 5multiples of 2 3HCF of 3 and 6 4, 6, 8prime numbers 1, 2, 3, 5, 6, 10, 15, 301 Highest Common FactorHCF HCF of 15 and 162. Write true or false for each of the following statements:(a) The HCF of 2 consecutive odd number is 2.(b) The HCF of 2 consecutive even number is 2.(c) The HCF of an even and an odd numbers is even.(d) The HCF of 2 coprime number is 1.(e) The HCF of 2 distinct prime number is 1.3. Find the HCF of the following by using prime factorization method: (a) 12, 18 (b) 32, 56 (c) 24, 88 (d) 35, 80Exercise 2.41. Find the HCF of the following by using definition method: (a) 18, 48 (b) 78, 112 (c) 36, 72, 90 (d) 32, 48, 144(e) 15, 30, 45 (f) 32, 48, 72 (g) 36, 144, 216 (h) 35, 70, 1052. Find the HCF of the following by using prime factorization method: (a) 26, 78 (b) 126, 144 (c) 150, 375 (d) 125, 625(e) 50, 100 (f) 18, 24, 30 (g) 20, 24, 60 (h) 21, 35, 56(i) 30, 40, 45 (j) 72, 162, 198 (k) 60, 75, 90 (l) 60, 100, 140 3. Find the HCF of the following by using division method: (a) 54, 72 (b) 121, 143 (c) 144, 168 (d) 312, 384

Acme Mathematics 7 45(e) 384, 432 (f) 437, 779 (g) 403, 527 (h) 620, 2108(i) 625, 1225 (j) 21, 28, 35 (k) 17, 51, 68 (l) 20, 60, 80(m) 15, 35, 120 (n) 54, 72, 108 (o) 90, 125, 342 (p) 144, 384 , 432 (q) 532, 931,1463 (r) 1617, 2871, 4257 4. Find the greatest number of boys among whom 345 oranges and 280 bananas can be equally divided? 5. Among how many students can 75 bananas, 100 oranges and 125 apples be equally distributed? What will be the share of each student? 6. Find the greatest number which exactly divides 72, 180 and 276. 7. Find the largest number which exactly divides 64, 288 and 352. 8. Find the greatest number that divides 532, 1188 and 1452 without remainder. 9. What is the greatest length which can be used to measure exactly 7 m 20 cm, 12 m and 36 m? 10. A room measures 738 cm by 414 cm. Find the size of the largest possible square tile that can be used to pave it. Also find the number of tiles. 11. Find the greatest number which divides 159 and 198 leaving the remainder 3 in each case. 12. Find the greatest number by which 323, 539 and 827 can be divided so as to leave a remainder 35 in each case. 13. Among how many maximum number of students can 105 books, 156 copies and 207 pens be equally divided leaving 7 books, 2 copies and 11 pens? 14. Find three numbers between 1500 and 2000 whose HCF is 120. 15. What should be added to 43 so that the sum is the HCF of 255, 357 and 459?16. What should be subtracted from 38 so that difference is the HCF of 532, 700 and 840?17. In the Covid period 1617 kg 'Daal', 4257 kg rice and 2871 kg potato were distributed among the group of families. At most how many families does it can be distributed and how much kg of each item each family will get?

46 Acme Mathematics 72. Lowest Common Multiple (LCM)In class six, we learnt to find out LCM of numbers by prime factorization method. Now, in this class we discuss about different methods to find the LCM of given numbers. Consider the following example: Set of multiples of 4, M4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, ...} Set of multiples of 6, M6 = {6, 12, 18, 24, 30, 36, 42, ...} Set of common multiples of 4 and 6 = {12, 24, 36, ...} Lowest common multiple of 4 and 6 = 12. ∴ LCM of 4 and 6 is 12. Thus, the least common multiple of two or more than two numbers is called the lowest common multiple (LCM) of those numbers. The given numbers exactly divides the LCM.We will now learn three methods to find the HCF of a given number. These methods are : (a) LCM by Definition Method Consider three numbers 2, 3 and 4. The multiples of 2, 3 and 4 can be obtained as follows: The set of multiples of 2 = {2, 4, 6, 8, 10, 12, 14, 16, 18 , 20, 22, 24,……….} The set of multiples of 3 = {3, 6, 9, 12, 15, 18, 21, 24,…………}The set of multiples of 4 = {4, 8, 12, 16, 20, 24, ................ } The set of common multiples of 2, 3 and 4 = {12, 24, 36, ...... } Out of set of common multiples of 2, 3 and 4, 12 is the least. Thus, the LCM of 2, 3, and 4 is 12. Note: The number 0 is excluded in the multiples while calculating the LCM.Solved ExampleExample 1 Find the LCM of 6, 12 and 18 using definition method. Solution: Set of multiples of 6, M6 = {6, 12, 18, 24, 30, 36 , 42, 48, 54, 60, 66, 72 , ...} Set of multiples of 12, M12 = {12, 24, 36 , 48, 60, 72 , ...} Set of multiples of 18, M18 = {18, 36 , 54, 72 , 90, ...} Now, set of common multiple of 6, 12 and 18 = {36, 72, ....} The least common multiple is 36. Therefore, LCM of 6, 12 and 18 is 36.

Acme Mathematics 7 47(b) LCM by Prime Factorization Method In this method we factorize each of the given numbers into many prime numbers and put the numbers in the product of all the prime numbers. Then we find the product of all prime numbers which may or may not be common to each of the given numbers. This product is the LCM. Example 2 Find the LCM of 6, 12 and 18 using prime factorization method. Solution: Here, 6 = 2 × 3 12 = 2 × 2 × 3 18 = 2 × 3 × 3 Common factor of 6, 12 and 18 = 2 × 3 = 6 Remaining factor of 6, 12 and 18 = 2 × 3 = 6 Lowest common multiple = Common factor × Remaining factors = 6 × 6 = 36 ∴ LCM of 6, 12 and 18 is 36. (c) LCM by Division Method In this method we divide at least two number by the smallest common prime factors and the same process is repeated until division is possible. The product of all the divisors and remaining numbers is the LCM of given numbers. Consider the numbers; 6, 12 and 182 6, 12, 183 3, 6, 91, 2, 3Hence, 2 and 3 are divisor 1, 2 and 3 are remaining numbers.∴ LCM =Solved ExampleExample 1 Find the smallest number divisible by each of the number 12, 18 and 24. Solution: The smallest number divisible by 12, 18 and 24 is their LCM.Now,12 = 2 × 2 × 318 = 2 × 3 × 324 = 2 × 2 × 2 × 3Here, LCM = 2 × 2 × 3 × 3 × 2 = 72 Therefore, the required smallest number is 72. 2 632 122 632 183 932 × 3 × 2 × 3 × 1 divisor= 6 × 6 = 36 remaining numbers

48 Acme Mathematics 7Example 2 Two bells ring at the intervals of 8 and 12 minutes respectively. At what time do they ring together, if they start ringing simultaneously at 12:00 noon? Solution: Required time is the LCM of 8 minutes and 12 minutes.Now,2 8, 122 4, 62, 3LCM of 8 and 12 = 2 × 2 × 2 × 3 = 24, [24 minutes] Therefore, two bells ring together at (12:00 + 0:24) = 12:24 P.M. Example 3 Find the least number which when divided by 12, 15 and 18, will leave 6 remainder in each case. Solution: The required number is LCM + 6.2 12, 15, 183 6, 15, 92, 5, 3LCM = 2 × 3 × 2 × 5 × 3 = 180 Required number = 180 + 6 = 186 Hence, 186 is the least number which when divided by 12, 15 and 18 will leave 6 remainder in each case.Classwork1. Write true or false for the following statements:(a) A factor of a number divides the number exactly.(b) A multiple of a number is exactly divisible by the number.(c) Every number is a factor as well as a multiple of itself.(d) 2 is the only even prime number.(e) 1 is a factor of every number.(f) 1 is neither prime nor composite number.(g) The product of HCF and LCM of 2 numbers equal their product.(h) Coprime numbers have 1 as their only common factor.2. Find the LCM of the following numbers by using definition method: (a) 3, 5, 6 (b) 4, 6, 9 (c) 8, 12, 16 (d) 6, 9, 12

Acme Mathematics 7 49Exercise 2.51. Find the LCM of the following numbers by using definition method:(a) 10, 12, 15 (g) 12 , 16 (c) 10, 15 (d) 20, 50 (e) 40, 60 (f) 100, 125 (g) 80, 90 (h) 150, 200 2. Find the LCM of the following number by using prime factorization method: (a) 24, 36 (b) 15, 35(c) 30, 45 (d) 10, 75 (e) 40, 60 (f) 72, 108(g) 15, 30, 45 (h) 32, 48, 64(i) 75, 105, 150 (j) 64, 80, 160 3. Find the LCM of the following by using division method:(a) 12, 18, 20 (b) 20, 24, 30(c) 21, 63, 70 (d) 116, 120, 232 (e) 144, 190, 2884. Find the least number which is exactly divisible by 65 and 78. 5. Find the least number which is exactly divisible by 20, 28 and 42. 6. Find the least number which, when divided by 25, 30 and 45, will leave 10 remainder in each case. 7. Find the least number which, when divided by 75, 100 and 125 gives the same remainder 50 in each case. 8. What is the smallest number which, when increased by 3, is divisible by 21, 25 and 27? 9. Find the least number which, being increased by 10, will be exactly divisible by 24, 27 and 30. 10. Find the least number which, when decreased by 7, is exactly divisible by 45, 60 and 72. 11. Find the least number which, when decreased by 10, is exactly divisible by 75,100 and 50. 12. Three clocks ring at intervals of 10, 12 and 15 minutes respectively. If they all ring together at 10 a.m., at what time will they ring together again?

50 Acme Mathematics 713. Five bells ring after an interval of 5, 6, 7, 8 and 10 seconds. After what least interval of time do they ring together again? 14. The number of students in different classes in a hostel is given below:Classes 8 9 10Number of students 12 18 20(a) At least how many 'Samosas' are needed to distribute equally among them? (b) How many 'Samosas' will each class get? 15. How many numbers are there between 154 to 200 which are divisible by 2, 3 and 6? 16. Pemba has some stamps. If he pastes them in rows of 12, 15 or 18, no stamps will be left behind. What is the least possible number of stamps that he has?17. Look at these gears:(a) A complete 6 turns how many turns does B complete?(b) What is the least number of turns that A can complete so that B also completes an exact numbers of turns?(c) B completes 24 turns. On how many occasions will both A and B have been back in their starting position at the same time.(Hints: Count their pick parts)18. Bishnu's 3 children, Isha, Aayush and Aasutosh visit him on Sunday morning. Isha visits every 2 weeks. Aayush visits every 3 weeks and Aasutosh visits every 4 weeks. All three children visited last Sunday:(a) How often do Isha and Aauush visit on the same day?(b) How often do Isha and Aasutosh visit on the same day?(c) How often do Aayush and Aasutosh visit on the same day?(d) When they meet their father again in the same day?Project WorkTake three scales (ruler) 15 cm long, 30 cm long and 60 cm long. By using these three scales verify that their LCM is 60 cm.AB