Acme Mathematics 6

Example : Find the LCM of 4 and 8The lowest common multiple of two or more numbers is the smallest number which is divisible by each one of the given numbers.We use the following methods to find the LCM.(i) Set of multiples (ii) Prime factorization (iii) DivisionActivityExpress 15 and 35 in the diagram and find HCF and LCM.Here, 3 1555 357Therefore, 15 = 3 × 5 and 35 = 5 × 7Diagram:3 5 7(a) Common factor is 5. Hence, HCF = 5(b) Remaining factors are: 3 × 7Hence, LCM = Common factors × Remaining factors= 5 × 3 × 7 = 105Solved ExampleExample 1 : Find the LCM of 6 and 8.Solution: In prime factors,6 = 2 × 38 = 2 × 2 × 2Here, 6 and 8 have a common prime factor 2.It is only counted once. Other factors are 2, 2 and 3.∴ LCM = 2 × 3 × 2 × 2 = 24224,2,1,8 Dividing 4 and 8 by 2.Dividing 2 and 4 by 2.LCM = 2 × 2 × 2 = 842Acme Mathematics 6 51

Example 2 : Find the LCM of 2 and 3 by listing the set of multiples.Solution: Here,Set of multiples of 2, M2 = {2, 4, 6, 8, 10, 12, 14, 16, 18, .......}Set of multiples of 3, M3 = { 3, 6, 9, 12, 15, 18, .........}Set of common multiples = { 6, 12, 18, .......} Lowest common multiple = 6Hence, the LCM of 2 and 3 is 6.Example 3 : Find the LCM of 20 and 24 using prime factorization method.Solution: Here, 20 = 2 × 2 × 524 = 2× 2 × 2 × 3L.C.M. = 2 × 2 × 5 × 2 × 3 = 120Example 4 : Find the LCM of the numbers 15 and 20. Use division method.Solution: Here, LCM = 5 × 3 × 4 = 60Example 5 : Find the smallest number which when divided by 24 and 36 leaves no remainder.Solution: Here, the smallest number is the LCM of 24 and 36.Now, 24 = 2 × 2 × 2 × 336 = 2 × 2 × 3 × 3L.C.M. of 24 and 36 = 2 × 2 × 3 × 2× 3 = 72.Hence, the required number is 72. Classwork1. State true or false.(a) Every number is multiple of itself. (c) 1 is the smallest multiple number.(b) 2 is a multiple of 1. (d) All even numbers are multiple of 2.2. (a) List all the multiples of 2 between 1 and 10.(b) List all the multiples of 3 between 10 and 30.2 202 1052 242 122 63 common factors remaining factorsRemember! While dividing, at least two number must be divided.5 15, 203, 42 242 122 632 362 183 93 52 Acme Mathematics 6

(c) List the first 10 multiples of 5.(d) List the first 8 multiples of 7 below 60.3. List the first 8 multiples of the following numbers:(a) 2 (b) 3 (c) 4 (d) 5(e) 6 (f) 7 (g) 8 (h) 9 Exercise 2.61. Find the LCM of the following numbers by listing the multiples:(a) 2 and 4 (b) 4 and 6 (c) 4 and 8 (d) 6 and 10(e) 7 and 14 (f) 2 and 6 (g) 3 and 9 (h) 6 and 92. Write the multiples of the given pair of number and find their LCM.(a) 2, 3 (b) 3, 4 (c) 3, 5 (d) 4, 6 (e) 6, 8 (f) 5, 83. Find the LCM by using division method:(a) 6, 9 (b) 8, 10 (c) 9, 12 (d) 12, 15(e) 10, 15 (f) 14, 18 (g) 16, 24 (h) 20, 304. Find the LCM of the following numbers using prime factorization method.(a) 42, 72 (b) 36, 48 (c) 56, 105(d) 39, 52, 65 (e) 48, 60, 84 (f) 115, 130, 655. Find the LCM of the following numbers using division method.(a) 14, 20, 24 (b) 30, 45, 36 (c) 55, 95, 35(d) 14, 35, 56 (e) 36, 100, 144 (f) 100, 175, 210(g) 28, 35, 42 (h) 48, 132, 96 (i) 40, 75, 256. (a) Find the least number which is exactly divisible by 90 and 132. (b) Find the lowest number which is exactly divisible by 45 and 90. (c) Which is the smallest number that can be divided by 24 and 28 leaving no remainder. (d) Study the diagram. The step of Hari is 70 cm. The steps of Sita and Rita are 90 cm. How far is B from A ?7. (a) In our school the duration of a class in the primary section is 45 minutes and in the pre-primary section it is 60 minutes. If both sections start the first bell at 10 am when will the two bells ring together again ? A BAcme Mathematics 6 53

(b) Two bells ring at intervals of 15 minutes and 10 minutes respectively. They ring 10 am first. At what time will they ring together again ?8. Two numbers 20 and 25 are given:(a) Find the product of 20 and 25. (b) Calculate the HCF of 20 and 25.(c) Calculate the LCM of 20 and 25. (d) Find the product of HCF and LCM.(e) Compare the result of number (a) and number (d).(f) Write your conclusion.Project 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. Now, 1. List the number which is divisible by 2. 2. List the number which is divisible by 3. 3. List the number which is divisible by 5. After completing the listing, present it to your class room.H. Square and square rootA number written a little raised next to a number is the power of that number. It is also called the index of the component.For example : 43, Here, 4 is base number and 3 is called the power of 4.Meaning of 43 : 43 represents the multiplication of 4, three times by itself.So, 43 = 4 × 4 × 4 = 64Any number raised to the power 2 gives a square number. For example : 52 = 5 × 5 = 2572 = 7 × 7 = 4925 and 49 are square numbers.In another word, if a number has 2 identical factors, then the number is square number.43It is power.It is base.52 is read as 5 square.54 Acme Mathematics 6

Now, look at the table.Number Product (Number × Number) Square number1 1 × 1 12 2 × 2 43 3 × 3 94 4 × 4 165 5 × 5 256 6 × 6 367 7 × 7 498 8 × 8 649 9 × 9 8110 10 × 10 100Hence, first 10, square numbers are : 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100Square number can be illustrated by drawing squares.In the above drawing, number of room in each side is the square root of the total square rooms.Here, Square number = 52 = 5 × 5 = 25Number of rooms each side = 5∴ Square root of 25 = 5 is the square root sign. Taking the square root is the opposite of squaring.Thus, 25 = 5 × 5 = 525 5 × 5Square root sign Original numberSquare root= = 5Chess board is the good example of square 8 × 8 = 64Acme Mathematics 6 55

(a) Perfect square numbersStudy the given examples carefully.1 × 1 = 12 = 1, 1 is a perfect square number.2 × 2 = 22 = 4, 4 is a perfect square number.3 × 3= 32 = 9, 9 is a perfect square number.4 × 4 = 42 = 16, 16 is a perfect square number.5 × 5 = 52 = 25, 25 is a perfect square number.But,1 × 3 = 32 × 1 = 24 × 5 = 203, 2, and 20 are not a perfect square numbers.A perfect square number is the product of two same (identical) numbers.Prefect square number can write in the square shape from. 14 916 25(b) Square root of a perfect square numberConsider 25. It is represented by55In each side of the square there are 5 dots. But there are 25 dots in total. 5 is called the square root of 25.It is written as 25= 5or, 5 × 5 = 5or, 52 = 5Similarly, square root of 16 = 16 = 4 × 4 = 42 = 4Square root of 49 = 49 = 7 × 7 = 72 = 7I know, 32 is read as 3 square.the symbol ' ' is radical sign56 Acme Mathematics 6

Solved ExampleExample 1 : Find the square of 12.Solution: Here, given number = 12Now, square of 12 = 122= 12 × 12 = 144Example 2 : Find the square of 20.Solution: Here,Square of 20 = 202= 20 × 20= 400Example 3 : Find the square root of 225.Solution: Here, given number = 225Now, square root of 225 = 225 = 15 × 15 = 15Example 4 : Rs. 2025 was distributed as many rupees as there were students. How many students were there ?Solution: Here, 2025 is given number.Number of students is the square root of the number 2025.Now, 2025 = 5 × 5 × 3 × 3 × 3 × 3 = 5 × 3 × 3 × 5 × 3 × 3= 45 × 45= 45Square root of 2025 = 45Thus, there were 45 students.Example 5 : Find the square root of 64.Solution: Here,Square root of 64 = 64= 2 × 2 × 2 × 2 × 2 × 2 = 8 × 8 = 82 = 8Hence, square root of 64 is 8.Oh! for 20 × 20 multiply 2 × 2 and add 2 zeros so, easy.2 642 322 162228 4 Acme Mathematics 6 57

Example 6 : Simplify : 160000Solution: Here,160000 = 16 × 10000= 16 × 100 × 100 = 42 × 102 × 102 = 4 × 10 × 10 = 400Classwork1. Fill in the blanks.(a) 12 = .................... (b) 22 = .................... (c) 42 = ....................(d) 102 = .................... (e) 202 = .................... (f) 402 = ....................(g) 1002 = .................... (h) 2002 = .................... (i) 4002 = ....................2. Find out the following :(a) 72 (b) 102 (c) 152 (d) 202(e) 302 (f) 452 (g) 502 (h) 10023. Choose the square numbers.(a) 16 (b) 25 (c) 60 (d) 90(e) 120 (f) 144 (g) 225 (h) 400Exercise 2.71. Write correct number in the blanks.(a) 9 = 3 ............... (b) 4 = .............. (c) 16 = ...........(d) 100 = .................. (e) 400 = .............. (f) 1600 = ............(g) 10000 = .................. (h) 40000 = .............. (i) 160000 = ...........2. Find the square of these numbers.(a) 8 (b) 17 (c) 20 (d) 29(e) 53 (f) 65 (g) 80 (h) 993. Calculate the square root of the following numbers.(a) 4 (b) 16 (c) 49 (d) 25(e) 64 (f) 81 (g) 100 (h) 4 × 1004. By what least number should the following numbers be multiplied to make it a perfect square number.(a) 8 (b) 12 (c) 20 (d) 1058 Acme Mathematics 6

5. (a) A certain number of students collected Rs 81 for Red- cross. Every students gave as many rupees as they are. Find the number of students.(b) A square field has equal number of rows and columns of plants. If there are 100 plants in all find the number of plants in each row. (c) In our hall, there is a capacity for 64 people. The chairs are arranged in rows and columns in equal number, find the number of chairs in each row.(d) The area of a square ground is 49 m2. Find the length of side of the ground.(e) A hostel warden arranged the hostel students in square shape. In each line there are 9 students and 6 are left. Find(i) How many students were there ?(ii) How many students at least should be added to make again square shape ?6. What number multiplied by itself will give the product 121 ?7. A chessboard is a large square divided into 64 smaller squares. How many squares has it along each side ?8. An army commander orders his soldiers to stand in square form . He finds that there are 7 soldiers in the first row of the square and 6 are not in the line. Find the total number of soldiers .Project WorkWrite the numbers from 1 to 100 and make a chart and do the following activities. (a) colour the numbers red which are square number. (b) colour the numbers yellow which are cube number. After completing the colouring, present it to your class room.Acme Mathematics 6 59

1. (a) Define the prime number.(b) Find the sum of square of 8 and 12.(c) Write the largest number and the smallest number by using 7, 4, 6 and find their difference.2. (a) Define factors.(b) Simplify : 16 – 8 {17 – (45 ÷ 3)}(c) Show the factor of 120 by making tree diagram.3. (a) Write the set of factors of 20, F20(b) Find the HCF of 20 and 30 by prime factorization method.(c) Write the set of multiples of 6 between 40 and 80.4. (a) If n1 and n2 are two prime numbers, what would be it's HCF.(b) Simplify : 18 ÷ 6 + [8 {12 ÷ 2 of 3} – 4] – 5(c) Find the LCM of 20 and 25 by prime factorization method.5. (a) The smallest number which is exactly divisible by the given numbers is called ............... of given numbers.(b) A square field has equal number of rows and columns of plants. If there are 1225 plants in all, find the number of plants in each row.(c) Sani paid a sum of Rs. 1000 to a shopkeeper for buying 6 exercise book at the rate of Rs. 100 and 5 pens at the rate of Rs. 15. How much will she receive in return?6. Among three digit number formed by using 5, 0 and 1.(a) Write the greatest number.(b) Write the smallest number.(c) Find the difference between the greatest number and the smallest number.Mixed Exercise60 Acme Mathematics 6

7. Difference between 25 and 9 is divided by 8 then multiplied by 2.(a) Write the mathematical statement of given problem.(b) Simplify the statement and find the result.(c) What should be done to make the result zero ?8. Sima has Rs. 400. She bought a book at Rs. 180. Her uncle gave her Rs. 200.(a) Write the mathematical statement of given problem.(b) How much money does Sima has now?(c) Is 895 exactly divisible by 5? Find out by divisibility test.9. There are the digits 3, 1 and 9(a) The smallest number formed from the given digits = (b) The greatest number formed from the given digits = (c) Find the sum and difference between the greatest and the smallest numbers formed from the digits 3, 1 and 9.10. (a) The greatest number which is exactly divisible by the given numbers is called ...... a given number.(b) The game teacher tried to arrange grade six students in equal number of rows and columns with 10/10 students in each row and column, but the number of students is less by 7. Now find out the total number of students is the class.(c) Anu paid a sum of Rs. 1000 to a shopkeeper for buying five exercise books at the rate of Rs 80, two pens at the rate of Rs. 50 and 5 table tennis ball at the rate of Rs. 15. How much will she receive in return?11. (a) Fill in the blanks.(i) 3 billions = .......... lakhs (ii) 1 crore = .............. millions.(b) 2004 is divisible by ............ .(c) Prime factors of 15 are ......................12. (a) Define composite numbers.(b) A book and a copy cost Rs. 60 and Rs. 45 respectively. Find the smallest sum of money with which exact number of books and copies can be purchased.(c) Seats are arranged in 32 rows and 32 column is a film hall. How many seats are there in the hall altogether?Acme Mathematics 6 61

2.2 IntegersA IntroductionWe have the set of numbers as,The set of Natural numbers, N = { 1, 2, 3, 4, 5, 6, 7, 8, 9,.........}The set of whole numbers, W = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 .....} Now,Take any two numbers from the set of whole number,W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 .....}Suppose, they are 8 and 4 then,8 – 4 = 4, 8 + 4 = 12, 8 × 4 = 32, 8 ÷ 4 = 2But, what is the result of ( 4 – 8) ?Similarly, 4 – 2 = 2, 11 – 7 = 4, 8 – 4 = 4What are the results of (2 – 4), (7 – 10) and (5 – 8) ?Hence, we have no any number in the set of whole numbers like 4 – 8, 2 – 4, 7 – 10 and 5 – 8 etc.Now, Consider the following pattern.8 – 1 = 7, 8 – 2 = 6, 8 – 3 = 5, 8 – 4 = 4 8 – 5 = 3,8 – 6 = 2, 8 – 7 = 1, 8 – 8 = 0, 8 – 9 = ? 8 – 10 = ?Since 8 – 9 = ? we need to extend the whole system to include the negative numbers, which are the image of positive numbers if mirror is at zero, see the number line. –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7– 1, – 2, – 3, – 4, – 5, – 6, and – 7 are the mirror image of 1, 2, 3, 4, 5, 6, and 7 respectively. – 1 is the image of 1 and called negative of 1, the opposite of 1 or minus 1. Similarly, – 2 is minus 2, – 3 is minus 3 etc.The system consisting of natural number, their negatives and the number zero is called the set of Integers.62 Acme Mathematics 6

The set of integers is denoted by Z where Z = { ....., – 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, ......}. The position of 0 is called the point of reference.The set of numbers right to 0 is the positive integers and denoted Z+where, Z+ = { 1, 2, 3, 4, 5,..............}.The set of numbers on the left to 0 is called the negative integers and denoted by Z–where, Z– = {–1, –2, –3, – 4, .............}.The number 0 is called the zero integer.Negative integers Zero integer Positive integers–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8On the number line, the movement from left to right is positive movement and movement from the right to left is negative movement. Consider two numbers 3 and 6.Now, 3 – 6 in a number line is given below.–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16From the above number line 3 – 6 is 3 units left from 0 which is an image of 3. Hence we denote it by – 3.Similarly, 2 – 3 = –1, 7 –10 = – 3, 5 – 8 = – 3 etc. Only in the set of Integers the subtraction is defined.Look at given examples carefully,Solved ExampleExample 1 : Add: 3 + 9 on the number line.Solution:–2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 153 912∴ 3 + 9 = 12Negative (–) Positive (+)+ 3– 6Acme Mathematics 6 63

Example 2 : Add: – 3 – 3 on the number line.Solution:–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3–3– 6– 3∴ – 3 – 3 = – 6Example 3 : Find the product (– 2) × (+ 3)Solution:(– 2) × (+ 3)3 timessame direction (left)2 unitsgo to left– 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4∴ (– 2) × (+ 3) = – 6Classwork1. Fill in the blanks: (a) – 6 + 4 = ......... (b) – 7 – 6 = ......... (c) 4 – 10 = .........(d) 8 – 4 = ......... (e) 11 – 19 = ......... (f) – 5 + 2 = .........2. Look at the number line and fill in the boxes:(a)– 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5(+ 2) + = (b) (– 4) + = – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5(c) × = 60 1 2 3 4 5 6 7 8 9 1064 Acme Mathematics 6

3. Simplify:(a) 4 + (– 3) (b) 5 – (– 4) (c) 15 + (– 5)(d) (– 5) – (+ 2) (e) 20 – (– 22) (f) (– 5) – (+ 4)(g) (– 10) + (– 11) (h) (– 17) + (– 12) (i) (– 16) + (– 20)(j) – 16 – (– 8) (k) (– 2) (– 10) (l) (– 8) – (– 8)Exercise 2.81. Write number opposite to each integers:(a) 2 (b) – 4 (c) – 10 (d) 8 (e) – 1002. How many integers are there between – 1 to 5 ? Write them. (Use number line)3. How many integers are there between + 5 to – 5 ? Write them (Use number line)4. How many integers are there between 1 to – 3 ? Write them. (Use number line)5. Compare and put the correct sign in each box.(a) 5 – 4 (b) – 6 – 6 (c) – 4 – 1 (d) – 5 – 8 (e) – 10 – 10 (f) 0 1 6. Fill in the box with correct digit:(a) 2 + = 0 (b) 1 – 4 = (c) + 7 = – 4 (d) + = 10(e) 9 – 16 = (f) + 9 = 07. Draw a number line for the given integers.(a) – 3, – 2, – 1 , 0 (b) 0, 1, 2, 3, 4, 5, 6(c) – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4 (d) – 12, – 11, .............. + 5(e) 11, 12, 13, 14, 15, 16 (f) 4, 3, 2, 1, 0, – 1, – 2, – 3, – 48. Add the following using number line:(a) 2 + 5 (b) – 5 – 2 (c) 10 + (– 5)(d) 7 – 10 (e) 6 – 11 (f) 20 – 5Acme Mathematics 6 65

9. Multiply the following using the number line:(a) 3 and 4 (b) 5 and 4 (c) 5 and 3(d) – 5 and 2 (e) – 6 and – 2 (f) – 6 and – 110. Arrange in the increasing order and in the decreasing order:(a) – 1, – 3, 0, – 4, – 2 (b) – 7, – 5, 1, 7, 2, 9 (c) – 6, – 4, –2, 0, 8, 311. Find the integers two units on the left from the given integers using the number line.(a) 6 (b) – 7 (c) 0 (d) – 2 (e) 5 (f) 1212. Find the integers three units right from:(a) 67 (b) – 8 (c) 0 (d) – 3 (e) 15 (f) – 2013. Complete the following addition table. (Rule: Column + row)+ – 18 – 17 – 16 0 16 17 18– 18– 170+ 17+ 1814. Complete the following multiplication table: (Rule: Column × row)× – 3 – 2 – 1 0 1 2 3–3– 20+ 1+ 266 Acme Mathematics 6

1. (a) Write any two negative integers. They are ……. and ……..(b) Arrange in increasing and decending order: –2, –1, 0, 4, –3. The order is….....(c) Show the integers in the number line: – 4 and 52. Observe the figure:A B–5 –4 –3 –2 –1 0 1 2 3 4 5 6(a) Fill in the blanks by appropriate symbol '>' or '<' in each of the following cases.(i) – 4 ......... 0 (ii) – 5 ......... – 4(iii) – 3 ......... – 4 (iv) – 1 ......... 1(b) Write the integers between – 3 and 3.(c) What is the total distance from A to B (in km).3. Given : 0, 7, – 3, 9, – 13, 32, 36(a) Write the image of 36.(b) Write the above integers in an increasing order.(c) Write the above integers in a decreasing order.4. (a) The product of (+ 2) and (– 3) is .(b) Insert the appropriate sign in : 9 8 – 5 = 12(c) What is the result, when 6 is added to the quotient of 30 divided by 5.5. (a) Write down the opposite integer of -5.(b) Which is greater + 3 or – 4?(c) Write down the given integers in accending order.+ 4, – 2, + 3, 0, – 1, – 3, + 5Mixed ExerciseAcme Mathematics 6 67

EvaluationTime: 67 minutes Full Marks: 281. (a) Define a whole number. [1] (b) Simplify : 35 –{15– (19 +5) ÷ 3} [2] (c) Re–write the numerals using commas according to international number system: 725836174 [1]2. (a) Write the smallest and the greatest numbers that can be formed by the digits 1, 4, 0,5. [2] (b) The greatest number which exactly divides the given numbers is called ............. [1] (c) In a school, there are 144 students. If the students are arranged in square form in assembly, how many students should be placed in each row? [2] 3. (a) If n1 and n2 are two prime numbers, what would its HCF? [1](b) Simplify: 18 – 6 + [8 { 6 ÷ 2 x 3}– 4]–5 [2] (c) Find the L.C.M.: 18, 24 and 36 [2] 4. (a) Find the difference between the largest and the smallest number formed by 0,1,4,6 and 7. [2](b) Write the integers between –3 and 3? [2](c) Re–arrange the integers in an ascending order. Also mention the largest and the smallest integers. –2, 0, –6, 4, –3 [2]5. Sima has Rs. 400. She bought a book at Rs. 185. Her uncle gave her Rs. 200 .(a) Write the mathematical statement of given problems. [1](b) How much money does Sima has now ? [1](c) Add using the number line. (–2) + (+7) [1] 6. (a) Is 7542 exactly divisible by 3 ? Find out by divisibility test. [2](b) Write the factor's of 24. [1](c) Find the maximum numbers of students for whom 60 books and 72 pens can be equally distributed. [2]68 Acme Mathematics 6

2.3 Fraction Warm Up Test1. Choose the fractions as 'proper, improper or mixed'.134310923 112916 123101257100502. Put the sign >, = or < in the blanks.(a) 49 .............. 69 (b) 1020 ............. 12 (c) 12 ................ 22(d) 13 .............. 14 (e) 27 ............... 314 (f) 24 .............. 8163. Add or subtract:(a) 45 + 65 + 15 (b) 39 + 63 (c) 76 + 15(d) 912 – 212 (e) 37 – 114 (f) 17 – 1114. Write in decimal number.(a) 610 = ................. (b) 1210 = .............. (c) 15100 = ...........(d) 12 = ................... (e) 510 = ............... (f) 24 = ...... .........5. Convert into fractions.(a) 0.4 = ............... (b) 1.2 = ................ (c) 2.7 = ..............(d) 1.03 = ............. (e) 0.04 = ............... (f) 1.22 = ............6. Add or subtract.(a) 0.7 + 0.2 (b) 0.9 + 1.7 (c) 0.3 + 0.9 + 1.4(d) 0.8 – 0.3 (e) 1.3 – 0.9 (f) 1.0 – 0.8(g) 1.2 + 1.09 + 0.2 (h) 1.009 – 0.07 (i) 2.0 – 0.009Acme Mathematics 6 69

A. Types of fractionRevision“ Fractions with the same denominators are like fractions.For example : 27 , 37 , 47 etc are like fractions.“ Fractions with different denominators are unlike fractionsFor example : 23 , 35 , 47 etc are unlike fractions.“ Fractions in which the numerator is greater than the denominator are called improper fractions.For example : 75 , 43 , 127 etc are improper fractions.“ Fraction in which the numerator is less than the denominator are called proper fractions.For example : 35 , 27 , 910 etc are proper fractions.“ Fraction which are combination of whole numbers and proper fractions are called mixed fraction.For example : 1 23 , 4 56 , 7 111 etc are mixed fractions.“ Fraction of the types 12 , 24 , 612 , 1020 etc. are equivalent fractions.B. Equivalent fractionLook at the given examples.12 is shaded24 is shaded48 is shadedAll shaded parts are equal.All fractions 12, 24 and 48 are equal in area.27numeratordenominator70 Acme Mathematics 6

Similarly, Look at the given figures carefully.24 is shaded 48 is shaded816 is shaded 1224 is shadedHere, the shaded parts represented by 24‚ 48 and 816 are equal in area. 24, 48 and 816 are equivalent fractions. But 12, 23 , 34 , 45 etc are not-equivalent fractions.If the fractions represent equal part of the whole then the fractions are called equivalent fractions.All fraction represent the equal shaded part.132639412All shaded parts are equal.13, 26, 39 and 412 are equal fractions.13, 26, 39 and 412 are called equivalent fractions.Test of equivalent fractionsAre the following fractions equivalent ?(i) 15 and 525 (ii) 27 and 34Here in (i) given fractions are 15 and 525.Now, cross products are 1 × 25 = 25 and 5 × 5 = 25. Since cross product are equal. The fractions 15 and 525 are equivalent.In (ii) given fractions are 27 and 34.Cross Product15525Acme Mathematics 6 71

Now, cross products are 2 × 4 = 8 and 7 × 3 = 21.Since cross products are not equal. The fractions 27 and 34 are not equivalent.How we test the equivalent fractions?Process to test the equivalent fractions,™ Find the cross products.™ If the cross products are equal, the fractions are equivalent.How to find equivalent fractions?Study the given fractions carefully.12 = 1 × 12 × 1 = 12 Multiplying by 1.12 = 1 × 22 × 2 = 24 Multiplying by 2.12 = 1 × 32 × 3 = 36 Multiplying by 3.Let's see more examples.Example 1 : Find 3 equivalent fraction of 23.Solution: Here,23 = 2 × 13 × 1 = 2323 = 2 × 23 × 2 = 4623 = 2 × 33 × 3 = 6923 = 2 × 43 × 4 = 812Classwork1. Shade the following fractions:(a) (b) (c)122436By multiplying numerator and denominator by the same number we get equivalent fractions.All are equivalent fractions.23, 46, 69, 812 are equivalent fractions.72 Acme Mathematics 6

2. Match the equivalent fractions:(a) 12 (i) 68(b) 34 (ii) 315(c) 15 (iii) 46(d) 23 (iv) 243. Match the following equivalent shaded parts:(a)(b)(c)(d)Exercise 2.91. Fill in the boxes and make equivalent fractions.(a) 12 = 4 (b) 23 = 6 (c) 15 = 15(d) 3 = 69 (e) 1 = 412 (f) 1020 = 22. Write the equivalent fraction for each of the following.(a) 34 with 16 as denominator. (b) 13 with 7 as numerator.Acme Mathematics 6 73

(c) 36 with 36 as denominator. (d) 49 with 36 as numerator.(e) 68 with 40 as denominator.3. Are the following equivalent ?(a) 210 and 315 (b) 35 and 512 (c) 34 and 1215(d) 67 and 2428 (e) 23 and 1827 (f) 59 and 25454. Find equivalent fractions for each of the following fractions.(a) 27 (b) 35 (c) 115 (d) 325(e) 910 (f) 78 (g) 56 (h) 135. Complete the following and shade:(a) (b) (c)6. Fill in the box:(a) 12 = 24 (b) 23 = 6(c) 15 = (d) 27 = 14(e) 34 = 12 (f) 29 = 67. Multiply by 2, 3, 4 and find the equivalent fractions of the given fractions.(a) 12 (b) 23 (c) 13 (d) 34 (e) 24 (f) 158. Find equivalent fractions for the following.Multiple of 1.Multiple of 3.2639412515618721 (a) 13 = 13141626312212Remember!Equivalent fraction374 Acme Mathematics 6

(b) 25 = It is easy way. Now, I can do it.(c) 35 = (d) 47 = (e) 59 = 9. (a) Divide 1224 by 2, 3, 4 and find the equivalent fractions:122412 ÷ 224 ÷ 2= 612, 12 ÷ 324 ÷ 3 = 48, 12 ÷ 424 ÷ 4 = 361224, 612, 48 and 36are equivalent fractions.(b) Divide 20100 by 2, 4, 5 and find the equivalent fractions:(c) Divide 6084 by 2, 3, 4 and find the equivalent fractions:(d) Divide 150250 by 2, 5, 25, 50 and find the equivalent fractions:C. Fractions in lowest termsFractions must always be expressed in their lowest term.A fraction can be reduced to its lowest terms by dividing the numerator and denominator by the same number.Consider the fraction 1215.Its numerator is 12. Its denominator is 15.We can write 12 = 2 × 2 × 315 = 3 × 5Oh! Division rule is also easy.I know 2, 2, 3, are factors of 12Acme Mathematics 6 75

Now, 1215 = 2 × 2 × 33 × 5 take away the common factors.= 2 × 25= 4545 is the lowest term of 1215.Solved ExampleExample 1 : Reduce 2028 to its lowest term.Solution: Here, 2028 = 2 × 2 × 52 × 2 × 7= 57Example 2 : Reduce 2030 to the lowest terms.Solution: Here, 2030= 2 × 2 × 53 × 2 × 5 = 23Hence, 23 is the lowest term of 2030D. Comparison of fractionsStudy the given fractions. 12 13 14 15Rough2 20 2 282 10 2 145 72 202 1052 303 15576 Acme Mathematics 6

12 is greater than 13, 13 is greater than 1414 is greater than 15Hence, 12 > 13 > 14 > 15To compare any two quantities, it is necessary that the quantities belong to the same measure. In case of fractions, two fractions must be like fractions. To convert any two unlike fractions to like fractions, we have to give their equivalent forms.Consider the fractions 12 and 13.1213It is not possible to compare one part of 2 parts with one part of 3 parts. So let us convert them to their equivalent fractions. We can compare the fractions by making denominator same. LCM of 2 and 3 is 6.Now, convert 12 and 13 to their equivalent fractions with 6 as the denominator.12 = 1 × 3 2 × 3 = 3613 = 1 × 2 3 × 2 = 2636 and 26 are like fractions.2636From the diagram 36 is greater than 26. i.e. 36 > 26Hence, 12 > 13.'>' indicates the greater than.Oh! I know.Smaller the denominator greater thefractions, if their numerator is same.Only like fractions can be comparedAcme Mathematics 6 77

Solved ExampleExample 1 : Convert 23 and 34 into like fractions.Solution: Here, 23 and 34 are given fractions.The LCM of denominators 3 and 4 = 3 × 4 = 12Now, 23 = 2 × 43 × 4 = 81234 = 3 × 34 × 3 = 912812 and 912 are like fractions as their denominator is same.Example 2 : Arrange 27, 57 and 47 in decreasing order.Solution: Here, 27, 57 and 47 are like fractions. So, 57 > 47 > 27.Hence, decreasing order is 57 > 47 > 27.Example 3 : Arrange 23, 12 and 16 in increasing order.Solution: Here, 23, 12 and 16 are unlike fractions.LCM of 3, 2 and 6 is 6.Now, 23 = 2 × 2 3 × 2 = 4612 = 1 × 3 2 × 3 = 3616 = 1 × 1 6 × 1 = 16 Thus like fractions for 23, 12 and 16 are 46, 36 and 16 respectively where 16 < 36 < 46 i.e. 16 < 12 < 23.Hence, increasing order is 16, 12, 23.denominator of 34denominator of 232 3, 2, 63 3, 1, 31, 1, 178 Acme Mathematics 6

Classwork1. Choose the like fractions from the given fractions.19 , 34 , 27 , 39 , 110 , 512 , 29 , 79 , 97 , 892. Choose the unlike fractions from the given fractions.23 , 17 , 48 , 910 , 14 , 12 , 25 , 673. Fill in the blanks 'like fractions' or 'unlike fractions'.(a) 14 and 34 .................... (b) 79 and 610 ....................(c) 37 and 27 .................... (d) 58 and 310 ....................4. Put the correct sign > or < in the box.(a) 2818 (b) 2939 (c) 6757(d) 7939 (e) 10121112 (f) 3444(g) 4749 (h) 21029 (i) 710712(j) 810813 (k) 310311 (l) 12151214Exercise 2.101. Reduce the following fractions to their lowest term.(a) 20100 (b) 27108 (c) 120144 (d) 8496(e) 112196 (f) 60182 (g) 150250 (h) 2083122. Covert the given unlike fractions into like fractions. (Change into same denominator)(a) 14 , 35 (b) 45 , 56 (c) 710 , 25(d) 34 , 45 (e) 56 , 78 (f) 27 , 243. Find the smaller fraction in each pair.(a) 25 or 35 (b) 34 or 45 (c) 35 or 12(d) 710 or 520 (e) 47 or 514 (f) 15 or 415Acme Mathematics 6 79

4. Find the greater fraction in each pair.(a) 47 or 37 (b) 611 or 1012 (c) 10 21 or 13 (d) 25 or 26 (e) 34 or 1016 (f) 15 or 4155. Use the sign in the blank >, = or <.(a) 26 ............. 16 (b) 913 .............. 1213 (c) 79 ............... 49(d) 35 ............. 610 (e) 13 ................. 14 (f) 25 ................ 26(g) 12 .............. 100200 (h) 412 ............... 26 (i) 10 12 .............. 456. Arrange the following fractions in increasing order.(a) 25 , 28 , 27 , 210 , 23(b) 35 , 36 , 37 , 39 , 310(c) 18 , 68 , 58 , 38 , 28(d) 27 , 37 , 17 , 67 , 577. Arrange the following fractions in decreasing order.(a) 36 , 38 , 36 , 34 , 35(b) 517 , 59 , 57 , 512 , 515(c) 17 , 37 , 67 , 47 , 27(d) 29 , 69 , 19 , 79 , 398. Arrange the following fractions in: (i) Increasing order (ii) decreasing order(a) 25, 34, 1120 (b) 12, 13, 14 (c) 16, 29, 518(d) 514 , 47, 2028 (e) 49, 714 , 23 (f) 23, 34, 589. 34 of a stick is painted black, 12 is painted white and 16 is painted red. Which part is painted the most ?10. Narayan spends 13 of his income on food, 25 on house rent. Which amount is less?80 Acme Mathematics 6

E. Addition of fractions(a) Addition of like fractionsWhile adding like fractions, add the numerators. The denominator remains the same. Solved ExampleExample 1 : Add: 37 and 27.Solution: Here, 37 + 27= 3 + 27= 57(b) Addition of unlike fractionsIn order to add fractions having different denominator, first we have to make the fraction have a common denominator. The common denominator is the LCM of the denominators.Solved ExampleExample 1 : Add : 25 + 310Solution : Here, the denominators are not same. So, we need to find equivalent fraction with same denominator.The common (same) denominator is 10.Now, 25 + 310= 2 × 25 × 2 + 310= 410 + 310= 4 + 3 10= 710Therefore 25 + 310 = 710Example 2 : Add : 35 + 47Solution : Here, denominators are different so, we take the LCM of 5 and 7.37275725 and 310 are unlike fractionsAcme Mathematics 6 81

LCM of 5 and 7 = 5 × 7 = 35Now, 35 + 47= 3 × 75 × 7 + 4 × 57 × 5= 2135 + 2035 = 21 + 2035 = 4135Example 3 : Add 2 14 + 1 25Solution : Here, 2 14 and 125 are mixed fractions. So, these can also be written as.2 + 14 + 1 + 24= (2 + 1) + 14 + 25= 3 + 1 × 5 4 × 5 + 2 × 45 × 4= 3 + 520 + 820= 3 + 5 + 820= 3 + 1320 = 31320Therefore, 214 + 125 = 31320Example 4 : Add: 23 + 1423 ⇒1⇒ 4Solution: Now, LCM of 3 and 4 is 12.So, 23 = 812 →14 = 312 →23 + 14 →= 812 + 312 = 1112It is common denominators for new fractions20 is the LCM of 4 and 5.82 Acme Mathematics 6

Example 5 : Add: 12 + 25 + 23Solution: Here, denominator are 2, 5 and 3. LCM of 2, 5 and 3 is 30.Now, 12 + 25 + 23= 1 × 15 2 × 15 + 2 × 6 5 × 6 + 2 × 103×10= 1530 + 1230 + 2030= 15 + 12 + 20 30= 47 30= 117 30Method - II12 + 25 + 23= 1 × 15 + 2 × 6 + 2 × 10 30 = 15 + 12 + 20 30 = 47 30= 117 30Example 6 : Simplify: 1 1 10 + 23 5 + 4 1 20Solution: 1 1 10 + 23 5 + 4 1 20= 11 10 + 13 5 + 8120= 11 × 2 + 13 × 4 + 81 × 1 20= 22 + 52 + 81 20= 155 20 = 715 20It is LCM of 2, 5, and 3.It is LCM of 10, 5, and 20.Acme Mathematics 6 83

Classwork1. Fill in the blanks.(a) 12 + 12 = ................ (b) 13 + 13 = ................ (c) 25 + 15 = ................(d) 68 + 18 = ................2. Find the sum or difference:(a) 1 + 2 3 (b) 2 + 1 2 (c) 3 - 1 4Exercise 2.111. Add the following:(a) 3 4 + 5 8 (b) 2 9 + 1 3 (c) 5 21 + 2 7(d) 3 5 + 14 20 (e) 9 12 + 1 24 (f) 4 6 + 1 18(g) 31 3 + 1 1 18 (h) 1 1 12 + 2 3 14 (i) 112 3 + 71 6(j) 11 5 + 2 1 30 (k) 11 5 + 23 4 (l) 12 5 + 21 72. Simplify the following:(a) 5 9 + 2 9 + 1 9 (b) 2 12 + 5 12 + 3 12 (c) 7 15 + 3 15 + 1 15(d) 12 + 25 + 34 (e) 512 + 16 + 23 (f) 12 + 27 + 1(g) 2 311 + 419 + 13 (h) 1 920 + 1 715 + 123 (i) 1123 + 116 + 1143. Manju gave two-fifth of a chocolate to Kundan and one-fourth to Kushal and ate the rest. How much of the chocolate did Kundan and Kushal eat ? [Hints: Whole chocolate = 1]4. Kripa planted one-fifth of her garden in the morning and two-seventh of it in the evening. What fraction of her garden was planted in one day ?84 Acme Mathematics 6

F. Subtraction of fractions(a) Subtraction of like fractionsWhile subtracting like fractions, subtract the numerators. The denominator remains the same.Solved ExampleExample 1 : Subtract: 47 – 37Solution: Here, 47 – 37= 4 – 3 7 = 17So(b) Subtraction of unlike fractionsIn order to subtract fractions having different denominator, first we have to make the fraction have a common denominator. The common denominator is the LCM of the denominators.Solved ExampleExample 1 : Subtract 56 – 25Solution: Here, 6 and 5 are the denominator. LCM of 6 and 5 is 30.Now, 56 – 25= 5 × 5 – 2 × 6 30= 25 – 12 30= 13 30Example 2 : Subtract : 35 – 110Solution: Here, denominators are not same but 10 is multiple of 5.So, 35 – 110= 3 × 25 × 2 – 110= 6 – 110= 510 4737172 times 5 is 10Acme Mathematics 6 85

Example 3 : Subtract: 310 – 27Solution: Here, denominators are not same.LCM of 10 and 7 = 70So, 310 – 27= 3 × 710 × 7 – 2 × 107 × 10= 2170 – 2070= 21 – 2070= 170Example 4 : Subtract: 367 – 135Solution: Here, fraction are mixed fractionsSo, 367 – 135= (3 – 1) + 67 – 35= 2 + 6 × 5 7 × 5 – 3 × 75 × 7= 2 + 30 35 – 2135= 2 + 30 – 2135= 2 + 935= 2 935Example 5 : Simplify: 17 + 37 – 27Solution: Here, 17 + 37 – 27= 1 + 3 – 27= 4 – 27= 2786 Acme Mathematics 6

Example 6 : In a school, 13 of the students are girls what fractions are boys ?Solution: The total (whole) students = 1 (let)Girls = 1 3Boys = whole students – girls= 1 – 13= 11 – 13= 1 × 31 × 3 – 13= 33 – 13= 3 – 13 = 23Thus, 23 of the students are boys.Classwork1. Fill in the blanks.(a) 10 12 – 112 = ............. (b) 19 20 – 920 = ............ (c) 79 – 29 = ..............(d) 29 30 – 21 30 = ................ (e) 3 4 – 1 4 = ................ (f) 6 8 – 4 8 = ................2. Find the difference:(a) 5 – 1 2 (b) 4 – 2 3 (c) 3 – 4 5Exercise 2.121. Subtract the following:(a) 710 – 310 (b) 25 9 – 12 9 (c) 23 8 – 11 8(d) 59 – 16 (e) 15 – 115 (f) 18 – 548(g) 23 8 – 15 8 (h) 52 7 – 32 7 (i) 12 3 10 – 10 1 10(j) 25 8 – 1 1 12 (k) 51 2 – 23 4 (l) 132 7 – 1 1 35Acme Mathematics 6 87

2. Simplify: (a) 1115 + 115 – 715 (b) 2025 – 325 + 725 (c) 127 + 227 – 37(d) 314 + 114 – 34 (e) 315 – 15 + 45 (f) 413 – 213 + 53(g) 5 113 – 2 113 – 113 (h) 3 315 – 2 215 – 1 115 (i) 6 150 – 2 150 – 1503. Simplify the following:(a) 3 512 – 449 + 313 (b) 9 715 – 712 + 45 (c) 1 920 + 2 715 – 123(d) 312 – 223 + 114 (e) 537 + 449 – 235 (f) 512 + 213 – 4154. Nitesh spent five-eighths of his money on books and one-fourth of his money on copies. What fraction of his money did he have now? [Hints: Total money = 1]5. Sanu buys a ribbon of length 213 m. Rina buys a ribbon of length 312 m. Who has longer ribbon and by how much ?6. A man bought a rope of length 8 m. He cut out two pieces, one of them was 312 m long and the other was 2 512 long. How much of the rope is left ?7. Ram completed 12 of the work and Hari, 13 of it. How much work is left now?8. Narayani bought a rope of length 15 meter. She cut out two pieces one of the length 612 m and the other of length 234 m. How much of the rope is left?9. In a school, 25 of the students are boys. What fraction are girls?10. Subtract 45 from the sum of 210 and 35 .11. Baburam bought 314 m of iron wire and 213 m of silver wire. Which wire is short? By how much?12. Kiran can do 25 of a work in 1 day and Puspa can do 110 of a work in 1 day. If they work together how much work would they do in 1 day ? Who works more?88 Acme Mathematics 6

G. Multiplication of FractionsActivityTry to multiply the fraction. Discuss about the product.Multiply : 13 × 12121612Folded vertically 12 folded again horizontally 13Double shaded part is the product of 13 and 12. It is 16.Thus, 13 × 12 = 1 × 13 × 2 = 16(a) Multiplication of a fraction by a whole numberStudy the example:Example 1 : Multiply 14 by 3.Solution : 14 × 3 = 3 × 14 = 3 times 14 The shaded part in each figure is 14 The total shaded parts = 14 + 14 + 14 = 3 times 14 = 34 Addition method:14 + 14 + 14 = 1+ 1 + 1 4 = 34 This can be done by a short method as,14 by 3 = 14 × 3= 1 × 3 4 = 34 14 14 14 Remember!The numerator of the fraction is multiplied by the multiplier. The denominator remains the same.It is multiplier.Acme Mathematics 6 89

(b) Multiplication of a fraction by a fractionMultiply 12 by 13 12 × 13 = half of 13 This is whole (l)13 is shaded.Shaded part is made half = 16 is shaded.The shaded part is 12 of 13 Hence, 12 of 13 = 12 × 13 = 16.Solved ExampleExample 2 : Multiply: 25 and 1015 Solution: 25 and 1015 = 25 × 1015 = 2 × 105 × 15 = 2 × 2 × 55 × 3 × 5 = 415 (c) Multiplication of a whole number by a fractional numberExample 3 : Multiply: 8 by 12 Solution: Consider the following 8 parts are shaded4 parts are shaded12 of 8 parts are shaded.Hence, 12 of 8 Also, 12 of 8= 12 × 8 = 1 × 8 2 = 82 = 4Multiplying 8 by 12 is equal to Dividing 8 by 2.12 × 13 → half of 13 ™ Multiply the numerators.™ Multiply the denominators.™ Simplify if needed.How we multiply fractions?90 Acme Mathematics 6

Study the following examples.12 × 2 = 1 × 2 2 = 22 = 116 × 6 = 1 × 6 6 = 66 = 1110 × 10 = 1 × 10 10 = 10 10 = 1Similarly,45 × 54 = 4 × 55 × 4 = 2020 = 1 57 × 75 = 5 × 77 × 5 = 3535 = 110 9 × 910 = 9090 = 1™ If the product of a fraction and a whole number is 1 then each other is the multiplicative inverse of the other.™ If the product of two fractions is 1 then each of the fractions is the multiplicative inverse of the other.Multiplicative InverseMultiplicative inverse is also called Reciprocal number. Hence, 1 2 is reciprocal of 2.Similarly, 4 5 is reciprocal of 5 4 and 5 4 is reciprocal of 4 5 .(d) Value of the fraction Solved ExampleExample 1 : Find: 35 of 20Solution: Here, 35 of 20 = 35 × 20= 3 × 205 = 12Thus, 35 of 20 is 12.Classwork1. Multiply:(a) 13 × 5 (b) 116 × 9 (c) 110 × 7 (d) 111 × 10I know! 2 is multiplicative inverse of 12 and vice versa.I also know. 45 and 54are multiplicative inverse of each other.'of ' changes to '×' signAcme Mathematics 6 91

(e) 25 × 4 (f) 37 × 14 (g) 910 × 9 (h) 25 × 12(i) 512 × 24 (j) 719 × 19 (k) 112 × 4 (l) 220 × 52. Find the product:(a) 34 × 12 (b) 47 × 35 (c) 67 × 15 (d) 27 × 35(e) 25 × 18 (f) 211 ×34 (g) 77 × 28 (h) 79 × 121(i) 49 × 38 (j) 58 × 815 (k) 45 × 1012 (l) 1549 × 1445Exercise 2.131. Multiply:(a) 312 and 45 (b) 435 and 123 (c) 5 110 and 317(d) 115 and 115 (e) 316 and 119 (f) 417 and 1252. Find the value of:(a) 12 of 50 (b) 13 of 90 (c) 14 of 100 (d) 15 of 75(e) 25 of Rs. 20 (f) 34 of Rs. 40 (g) 27 of 49 m (h) 910 of 400 litre(i) 47 of 70 m (j) 710 of 120 km (k) 56 of 420 kg (l) 16 of 600 kg(m) 313 of 3 (n) 515 of 10 (o) 627 of 14 (p) 589 of 273. Simplify:(a) 34 × 15 × 47 (b) 23 × 1213 × 14 (c) 34 × 67 × 79(d) 19 × 35 × 15 (e) 67 × 1418 × 47 (f) 59 × 67 × 78(g) 112 ×123 × 15 (h) 234 × 1 211 × 47 (i) 123 × 456 × 1124. Draw the figures to represent the following:(a) 17 × 3 (b) 23 × 4 (c) 56 × 392 Acme Mathematics 6

H. Division of Fractions(a) Multiplicative inverseStudy the following examples.15 × 5 = 1 × 55 = 116 × 6 = 1 × 66 = 117 × 7 = 1 × 77 = 1In each of the above cases the product is 1.15 is multiplicative inverse of 5.5 is multiplicative inverse of 15Multiplicative inverse is also called the reciprocal.Similarly, 23 × 32= 2 × 33 × 2= 11= 1So, 23 is reciprocal of 32 and 32 is reciprocal of 23.Division is the inverse of the multiplication.as 23 × 32 = 1, also, 23 ÷ 23 = 1× Sign is changed to ÷ sign32 is changed to 23Consider the another example7 × 17 = 7 ÷ 7 = 117 is reciprocal of 7.Multiply 1 by 17 is equal to divided 1 by 7.We can write15 × 5 = 5 × 15 Acme Mathematics 6 93

(b) Dividing a whole number by a fractionConsider 8 ÷ 1 2 If represent 8Then represent 1. and represent 12.We see that there are 16 pieces of 1 2 in 8. So, 8 ÷ 1 2 = 16Now, Study the following 8 ÷ 1 2 = 16Also, 8 × 2 1 = 16 2 1 is multiplicative inverse of 1 2 The above division can be shown by a number line.∴ 8 ÷ 1 2 = 16While dividing a whole number by a fraction we multiply the whole number by multiplicative inverse (reciprocal) of the fraction.(c) Dividing a fraction by a whole numberConsider 1 2 ÷ 4If represents 1.Now, represents 12 of the whole. Then, represents 14 of 12 and 18 of the whole.Thus, 1 2 ÷ 4 = 1 8 There are 16 halves0 1 2 3 4 5 6 7 8121212121212121212121212121212121294 Acme Mathematics 6

The above division can also be shown by a number line.0 121It is 18 of the wholeIt is 14 of 12Hence, 1 2 ÷ 4 = 1 8 This is also obtained by 1 2 × 1 4 = 1 × 1 2 × 4 = 1 8 Thus, 1 2 ÷ 4 = 1 2 of 1 4 = 1 2 × 1 4 = 1 × 1 2 × 4 = 1 8 Solved ExampleExample 1 : Divide 35 by 3.Solution: Here, 35 by 3 = 35 ÷ 3= 35 × 13= 3 × 1 5 × 3= 15Thus, 35 ÷ 3 = 15While dividing a fraction by a whole number (except 0) we multiply the fraction by the multiplicative inverse of the whole number.It is multiplicative inverse of 4.4 is adivisor.reciprocal of 3 is 13Acme Mathematics 6 95

(d) Dividing a fraction by a fractionConsider 3 4 ÷ 1 4 Here, we find that there are 3 pieces of 1 4 in 3 4 So, 3 4 ÷ 1 4 = 3 Also, 3 4 × 4 1 = 3Hence, 3 4 ÷ 1 4 = 3 4 × 4 1 = 3Solved ExampleExample 1 : Divide 31 2 by 1 2 Solution: Here, 31 2 = 7 2 Now, 7 2 ÷ 1 2 = 7 2 × 2 1 = 7 × 2 2 × 1 = 14 2 = 7Hence, 31 2 ÷ 1 2 = 7Example 2 : Divide: 1 23 by 234 Solution: Here, 1 23 ÷ 234 = 53 ÷ 114= 53 × 411= 5 × 43 × 11= 2033Thus, 1 23 ÷ 234 = 2033Classwork1. Write the multiplicative inverse of the following numbers:(a) 47 (b) 79 (c) 81 (d) 1015(e) 19 (f) 12 (g) 20 (h) 418 96 Acme Mathematics 6

2. Divide: (a) 79 ÷ 7 (b) 511 ÷ 5 (c) 14 ÷ 4 (d) 35 ÷ 8(e) 45 ÷ 7 (f) 910 ÷ 3 (g) 310 ÷ 6 (h) 27 ÷ 2(i) 59 ÷ 15 (j) 15 ÷ 3 (k) 56 ÷ 10 (l) 2025 ÷ 20Exercise 2.141. Divide:(a) 3 ÷1 2 (b) 10 ÷2 5 (c) 6 ÷3 5 (d) 3 ÷ 21 4 (e) 9 ÷3 4(f) 2 ÷1 3 (g) 3 7 ÷ 3 (h) 6 7 ÷ 5 (i) 5 6 ÷ 42. Divide:(a) 5 8 ÷ 5 4 (b) 15 16 ÷ 5 4 (c) 21 28 ÷ 7 14(d) 6 8 ÷ 3 4 (e) 12 7 ÷ 3 14 (f) 11 10 ÷ 10 113. Divide:(a) 27 by 214 (b) 56 by 1012 (c) 2120 by 380 (d) 1011 by 111 (e) 127 by 127 (f) 54 by 2520 (g) 413 by 133 (h) 412 by 1810 (i) 513 by 169 (j) 214 by 234 (k) 217 by 514 (l) 319 by 2829 4. Find the result:(a) 14 9 ÷ 3 4 (b) 35 9 ÷ 22 3 (c) 44 5 ÷ 2 2 15 (d) 24 7 ÷ 36 49 (e) 18 9 ÷ 210 12 (f) 32 7 ÷ 22 7 5. Simplify:(a) 6 8 × 2 6 × 4 7 (b) 7 12 × 24 14 ÷ 4 1 (c) 6 58 × 14 12 ÷ 88 29(d) 23 4 × 8 11 ÷ 18 9 (e) 34 5 ÷ 19 10 × 21 2 (f) 16 17 ÷ 8 9 × 1 2(g) 5 8 ÷ 5 8 ÷ 5 8 (h) 101 2 ÷ 2 1 10 × 44 5 (i) 3 4 × 19 21 ÷ 38 42 Acme Mathematics 6 97

I. Simplification of fractionsSimplification of fractions is similar as the other simplification. Simplification can be done by using four simple rules. The rules are addition, subtraction, multiplication and division with brackets. The example given below gives the clear idea about it.Solved ExampleExample 6 : 1 litre of milk costs Rs. 11012. What is the cost of 12 litre of milk ?Solution: Here, 1 litre of milk costs Rs. 1101212 litre of milk costs 12 × Rs. 11012= Rs. 12 × 2212= Rs. 1 × 2212 × 2 = Rs. 5514.Example 7 : In a class there are 42 girls. If the fraction of girls in the class is 710 , find total number of students in the class.Solution: Here, let total number of students = 1 710 of the students = girls710 × No. of students = 42No. of students = 42 × 107 = 6 × 10= 60Thus number of students in the class is 60.Classwork1. Solve the following word problems.Mrs. Sharma bought the following foods.Maida : 212 Sugar : 1 kg 12 Dal : kg 14 Rice : 3 kg 12 kg98 Acme Mathematics 6

(a) Find the total weight of her foods.(b) Calculate the total weight of 'Dal', 'Sugar' and 'Maida'.2. The parts of a stick are given. 314518518314(a) Calculate the length of coloured part.(b) Find the length of uncoloured part.(c) If the black coloured part is 20 cm, calculate the length of the stick.Exercise 2.151. Find the sum of 34 and 12 and multiply it by 8202. Subtract 23 from the product of 37 and 1415.3. Divide the product of 1017 and 2840 by 117 .4. Divide 1021 by 56 and multiply the quotient by 23 – 1 2 .5. Simplify:(a) 12 + 115 × 611 – 23 (b) 34 × 13 – 14 – 13(c) 45 × 17 ÷ 87 + 13 (d) 23 ÷ 14 ×35 + 12(e) 345 – 1 110 ÷ 11 5 + 1 110 (f) 712 ÷ 5 + 212 – 3 46. Study the given picture of a group of students.(a) How many students are there in the group?(b) Express number of boys in fraction.(c) Express number of girls in fraction.(d) Find the sum of fractions.(e) What does '1' represent in the fraction?Acme Mathematics 6 99

2.4 Decimal A. RevisionLook at the following blocks.It has 10 units.It represents 10 in base -10 system.The shaded part is 110.110 is written as 0.10.1 is a decimal number.Look at the following blocks.It has 100 units.It represents 100. The shaded part is 1100.1100 is written as 0.01.Similarly, 11000 is written as 0.001. 0.001 is read as thousandths.Here, 0.1, 0.01 and 0.001 are decimal numbers.0.1 is read as zero point one.0.01 is read as zero point zero one.0.001 is read as zero point zero zero one. Consider a number 12.123.12.123 is a decimal number. It can be shown in place value chart as follows.Tens Ones Tenths Hundredth Thousandths1 2 1 2 3This is read as twelve point one two there.0.1 is read as tenths0.01 is read as hundredth100 Acme Mathematics 6