MAGNOLIA-STUDENT-TEXTBOOK-MATHEMATICS-G05-Combine

MATHEMATICS TEXTBOOK 5 Name: ___________________________________ Section: ________________ Roll No.: _________ School: __________________________________

Contents 5Class Part 1 5 Multiplication 5.1 Multiply Large Numbers ............................................................................................ 50 6 Division 6.1 Divide Large Numbers............................................................................................... 57 6.2 Factors and Multiples ................................................................................................ 63 6.3 H.C.F. and L.C.M. ....................................................................................................... 70

Chapter Multiplication 5 Let Us Learn About • properties of multiplication. • multiplying 4-digit and 5-digit by 2-digit and 3-digit numbers. • finding the missing numbers in the given product. • o bserving patterns in multiplication of numbers. Concept 5.1: Multiply Large Numbers Think Pooja’s mother bought 1750 kg of rice for the whole year at the price of ` 48 per kilogram. She asked Pooja to check if the bill is correct. How do you think Pooja can check it? Recall We have already learnt how to multiply a 4-digit number by a 1-digit number. Let us recall the basic concepts of multiplication. Properties of Multiplication Identity Property: For any number ‘a’, a × 1 = 1 × a = a. 1 is called the multiplicative identity. For example, 213 × 1 = 1 × 213 = 213. 50

Zero Property: For any number ‘a’, a × 0 = 0 × a = 0. For example, 601 × 0 = 0 × 601 = 0. Commutative Property: If ‘a’ and ‘b’ are any two numbers, then a × b = b × a. For example, 25 × 7 = 175 = 7 × 25. Associative Property: If ‘a’, ‘b’ and ‘c’ are any three numbers, then a × (b × c) = (a × b) × c. For example, 3 × (4 × 5) = (3 × 4) × 5 3 × 20 = 12 × 5 60 = 60 Let us answer the following to revise the the multiplication of 4-digit numbers. a) Th H T O b) Th H T O c) Th H T O 3234 1274 4567 ×2 ×8 ×5 d) Th H T O e) Th H T O f) Th H T O 5674 3120 4372 ×3 ×4 ×8 & Remembering and Understanding Multiplication of large numbers is the same as multiplication of 4-digit or 5-digit numbers by 1-digit numbers. If an ‘x’-digit number is multiplied by a ‘y’-digit number, then their product is not more than a ‘(x + y)’- digit number. Let us solve some examples of multiplication of large numbers. Multiplication 51

Example 1: Find these products. a) 2519 × 34 b) 4625 × 17 Solution: a) T Th Th H T O b) T Th Th H T O 12 23 413 2519 4625 ×34 ×17 11 1 0 0 7 6 → 2519 × 4 ones 3 2 3 7 5 → 4625 × 7ones + 7 5 5 7 0 → 2519 × 3 tens +4 6 2 5 0 → 4625 × 1 tens 8 5 6 4 6 → 2519 × 34 7 8 6 2 5 → 4625 × 17 Example 2: Find the product of 3768 and 407. Solution: T L L T Th Th H T O Here we can skip 323 the step ‘3768 × 0’ but, add one more zero in 545 3768 tens place while ×407 multiplying by 1 hundreds digit. 2 6 3 7 6 → 3768 × 7 ones + 1 5 0 7 2 0 0 → 3768 × 4 hundreds 1 5 3 3 5 7 6 → 3768 × 407 Example 3: Estimate the number of digits in the product of 58265 and 73. Then multiply and verify your answer. Solution: The number of digits in the multiplicand 58265 is five. The number of digits in the multiplier 73 is two. Total number of digits is seven. Therefore, the product of 58265 and 73 should not have more than seven digits. 52

Example 4: T L L T Th Th H T O Solution: 5 143 2 11 5 8265 ×73 11 11 1 7 4 7 9 5 → 58265 × 3 ones + 4 0 7 8 5 5 0 → 58265 × 7 Tens 4 2 5 3 3 4 5 → 58265 × 73 The number of digits in the product 4253345 is 7. Hence, verified. Find the product of 24367 and 506. T L L T Th Th H T O 2 133 2 244 2 4367 ×506 1 1 4 6 2 0 2 → 24367 × 6 ones + 1 2 1 8 3 5 0 0 → 24367 × 5 hundreds 1 2 3 2 9 7 0 2 → 24367 × 506 Multiplication 53

Chapter Division 6 Let Us Learn About • dividing 5-digit by 1-digit and 2-digit numbers. • rules of divisibility • finding prime and composite numbers. • factors, multiples, H.C.F. and L.C.M. of numbers. • prime factorisation of numbers. Concept 6.1: Divide Large Numbers Think Pooja’s brother saved ` 12500 in two years. He saved an equal amount every month. Pooja wanted to find his savings per month. How do you think Pooja can find that? Recall In Class 4, we have learnt dividing a 4-digit number by a 1-digit number. Let us now revise this concept with a few example. Divide: a) 3165 ÷ 3 b) 5438 ÷ 6 c) 2947 ÷ 7 d) 7288 ÷ 4 e) 1085 ÷ 5 57

& Remembering and Understanding Dividing a 5-digit number by a 1-digit number is the same as dividing a 4-digit number by a 1-digit number. Example 1: Divide: a) 12465 ÷ 5 b) 76528 ÷ 4 Solution: a) 2493 b) 19132 )5 12465 )4 76528 −10 −4 24 36 − 20 − 36 46 05 − 45 − 04 15 12 − 15 − 12 0 08 −8 0 Let us now divide a 5-digit number by a 2-digit numbers. Example 2: Divide: 21809 ÷ 14 Solution: Write the dividend and the divisor as Divisor Dividend Steps Solved Solve these 14 21809 Step 1: Guess the quotient by )20 53174 dividing the two leftmost digits by 14 × 1 = 14 the divisor. Find the multiplication fact which 14 × 2 = 28 has the dividend and the divisor. 14 < 21 < 28 So,14 is the number to be subtracted from 21. 58

Steps Solved Solve these Step 2: Write the factor other than Write 1 in the quotient and )13 34567 the dividend and the divisor as 14 below 21, and subtract. the quotient. Then bring down the next number in the dividend. 1 14 21809 −14 78 Step 3: Repeat steps 1 and 2 until 1557 )15 45675 all the digits of the dividend are brought down. )14 21809 Stop the division when the − 14 remainder < divisor. 78 − 70 80 − 70 109 − 98 11 Step 4: Write the quotient and the Quotient = 1557 remainder. The remainder must Remainder = 11 always be less than the divisor. Checking for the correctness of division: We can check if our division is correct using a multiplication fact of the division. Step 1: Compare the remainder and the divisor. Step 2: Check if (Quotient × Divisor) + Remainder = Dividend Division 59

Let us now check if our division in example 2 is correct or not. Step 1: Remainder < Divisor Dividend = 21809 Divisor = 14 Step 2: (Quotient × Divisor) + Quotient = 1557 Remainder = Dividend Remainder = 11 11 < 14 (True) 1557 × 14 + 11 = 21809 21798 + 11 = 21809 21809 = 21809 (True) Note: 1) If remainder > divisor, the division is incorrect. 2) If (Quotient × Divisor) + Remainder is not equal to Dividend, the division is incorrect. 60

Concept 6.2: Factors and Multiples Think Pooja learnt to find factors of a given number using multiplication and division. She wants to know the name given to the product obtained when we multiply numbers by counting. Do you know the name given to such products? Recall The numbers that divide a given number exactly are called the factors of that number. In other words, the numbers, which when multiplied ,give a product are called the factors of the product. For example, in 12 × 9 = 108, the numbers 12 and 9 are called the factors of 108. The number 108 is called the product of 12 and 9. Division 63

Complete the multiplication table of 8. 8×1=8 8×2= 8×3= 8×4= 8 × 5 = 40 8 × 6 = 48 8×7= 8 × 8 = 64 8×9= 8 × 10 = & Remembering and Understanding The products obtained when a number is multiplied by 1, 2, 3, 4, 5 …. are called the multiples of that number. In a multiplication table, a number is multiplied by the numbers 1, 2, 3, 4, 5 and so on till 10. In the multiplication table of 8, the products obtained are 8, 16, 24, 32, 40 and so on till 80. These are called the first ten multiples of 8. Similarly, a) 2, 4, 6, 8, 10, 12 … are the multiples of 2. b) 5, 10, 15, 20, 25, 30… are the multiples of 5. Let us now find the factors of some numbers. Factors of numbers from 1 to 10: Number Factors Number of Number Factors Number of 1 1 factors factors 1 6 1, 2, 3, 6 4 7 1, 7 2 2 1, 2 2 8 4 9 1, 2, 4, 8 3 3 1, 3 2 10 1, 3, 9 4 4 1, 2, 4 3 1, 2, 5, 10 5 1, 5 2 From the given table, we observe that: 1) The number 1 has only one factor. 2) The numbers 2, 3, 5 and 7 have only two factors (1 and themselves) 3) The numbers 4, 6, 8, 9 and 10 have three or four factors (more than two factors). Note: 1) The numbers that have only two factors (1 and themselves) are called prime numbers 2) T he numbers that have more than two factors are called composite numbers. 3) The number 1 has only one factor. So, it is neither prime nor composite. 64

Sieve of Eratosthenes Eratosthenes was a Greek mathematician. He created the sieve of Eratosthenes, to find prime numbers between any two given numbers. Steps to find prime numbers between 1 and 100 using the sieve of Eratosthenes: Step 1: Prepare a grid of numbers from 1 to 100. Step 2: Cross out 1 as it is neither prime nor composite. Step 3: Circle 2 as it is the first prime number. Then cross out all the multiples of 2. Step 4: Circle 3 as it is the next prime number. Then cross out all the multiples of 3. Step 5: Circle 5 as it is the next prime number. Then cross out all the multiples of 5. Step 6: C ircle 7 as it is the next prime number. Then cross out all the multiples of 7. Continue this process till all the numbers between 1 and 100 are either circled or crossed out. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 The circled numbers are the prime numbers and the crossed out numbers are the composite numbers. Division 65

There are 25 prime numbers between 1 and 100. These are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97. Note: 1) All prime numbers (except 2) are odd. 2) 2 is the only even prime number. Example 9: Find the factors: a) 16 b) 40 Solution: a) T o find the factors of a given number, express it as a product of two numbers as shown: 16 = 1 × 16 =2×8 =4×4 Then write each factor only once. So, the factors of 16 are 1, 2, 4, 8 and 16. b) 40 = 1 × 40 = 2 × 20 = 4 × 10 =5×8 So, the factors of 40 are 1, 2, 4, 5, 8, 10, 20 and 40. Example 10: Find the common factors of 10 and 15. Solution: 10 = 1 × 10 and 10 = 2 × 5 So, the factors of 10 are 1, 2, 5 and 10. 15 = 1 × 15 and 15 = 3 × 5 So, the factors of 15 are 1, 3, 5 and 15. Therefore, the common factors of 10 and 15 are 1 and 5. We can find the factors of a number by multiplication or by division. Example 11: Find the factors of 30. Solution: Factors of 30 Using multiplication 1 × 30 = 30 2 × 15 = 30 3 × 10 = 30 66

5 × 6 = 30 The numbers multiplied to obtain the given number as the product are called its factors. So, the factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. Using division 30 ÷ 1 = 30 30 ÷ 2 = 15 30 ÷ 3 = 10 30 ÷ 5 = 6 The different quotients and divisors of the given number are its factors. So, the factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. Facts on Factors 1) 1 is the smallest factor of a number. 2) 1 is a factor of every number. 3) A number is the greatest factor of itself. 4) Every number is a factor of itself. 5) The factor of a number is less than or equal to the number itself. 6) Every number (other than 1) has at least two factors – 1 and the number itself. 7) The number of factors of a number is limited. Let us now find the multiples of some numbers. Example 12: Find the first six multiples: a) 9 b) 15 c) 20 Solution: The first six multiples of a number are the products when the number is multiplied by 1, 2, 3, 4, 5 and 6. a) 1 × 9 = 9, 2 × 9 = 18, 3 × 9 = 27, 4 × 9 = 36, 5 × 9 = 45, 6 × 9 = 54. So, the first six multiples of 9 are 9, 18, 27, 36, 45 and 54. Now, complete these: b) 1 × 15 = 15, ___ × ___ = ____ , ___ × ___ = ___ , ___ × ____ = ____, ____ × ___ = ____, _____ × _____ = ____. So, the first six multiples of 15 are ____ , ____ , ____ , ____ , ____ and ____. Division 67

c) 1 × 20 = 20, ___ × ___ = ____, ___ × ___ = ___ , ___ × ____ = ____, ____ × ___ = ____ , _____ × _____ = ____. So, the first six multiples of 20 are ____, ____ , ____ , ____ , ____ and ____. Example 13: Find three common multiples of 10 and 15. Solution: Multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80, 90,100,…. Multiples of 15 are 15, 30, 45, 60, 75, 90, 105,…. Therefore, the first three common multiples of 10 and 15 are 30, 60 and 90. Facts on Multiples 1) Every number is a multiple of itself. 2) Every number is a multiple of 1. 3) A number is the smallest multiple of itself. 4) The multiples of a number are greater than or equal to the number itself. 5) The number of multiples of a given number is unlimited. 6) The largest multiple of a number cannot be determined. 68

Concept 6.3: H.C.F. and L.C.M. Think Pooja now knows prime and composite numbers. She wants to know a simple way to find H.C.F. and L.C.M. of two numbers. Do you know any simple method for the same? Recall We have learnt about prime and composite numbers and the definitions of H.C.F. and L.C.M. We first find the factors of the given numbers. The highest common number among them gives the H.C.F. of the given numbers. Likewise, we can find the multiples of the given numbers. The least common among them gives the L.C.M. of the given numbers. Let us revise the concept by finding the common factors of the following pairs of numbers. a) 12, 9 b) 15, 10 c) 30, 12 d) 24, 16 e) 35, 21 f) 36, 54 & Remembering and Understanding Prime numbers have only 1 and themselves as their factors. Composite numbers have more than two factors. So, composite numbers can be expressed as the products of their prime numbers or composite numbers. For example, 5 = 1 × 5; 20 = 1× 20 9 = 1 × 9, = 2 × 10 = 3 × 3; =4×5 We can express all composite numbers as the products of prime factors. 70

Expressing a number as a product of prime numbers is called prime factorisation. To prime factorise a number, we use factor trees. Let us see a few examples to understand this better. Example 19: Prime factorise 36. Solution: To carry out the prime factorisation of 36, draw a factor tree as shown. Step 1: Express the given number as a product of two factors. One of these factors is the least number (other than 1) that can divide it. The second factor may be prime or composite. Step 2: If the second factor is a composite number, express it as a product of two factors. One of these factors is the least number (other than 1) that can divide it. The second factor may be prime or composite. Step 3: Repeat the process till the factors 36 Step 4: cannot be split further. In other words, repeat the process till the factors do 2 × 18 not have any common factor other × 9 than 1. 2 × 2 Then write the given number as the product of all the prime numbers. 2 × 2 × 3 × 3 Therefore, the prime factorisation of 36 is 2 × 2 × 3 × 3. Note: A factor tree must be drawn using a prime number as one of the factors of the number at each step. Example 20: Prime factorise 54. Solution: Prime factorisation of 54 using a factor tree: 54 2 × 27 2 × 3 × 9 2 × 3 × 3 × 3 Therefore, the prime factorisation of 54 is 2 × 3 × 3 × 3. Division 71

Contents 5Class Part 2 10 Fractions - II 10.1 Add and Subtract Mixed Fractions .......................................................................... 26 10.2 Multiply Fractions ....................................................................................................... 31 11 Decimals - I 11.3 Add and Subtract Decimals ..................................................................................... 55 12 Decimals - II 12.1 Multiply and Divide Decimals................................................................................... 61

Chapter Fractions - II 10 Let Us Learn About • the terms ‘mixed’, ‘proper’ and ‘improper’ fractions. • a dding and subtracting mixed fractions. • multiplying and dividing fractions by fractions. • finding the reciprocals of fractions. Concept 10.1: Add and Subtract Mixed Fractions Think Pooja has learnt addition and subtraction of unlike fractions. She has also learnt the conversion of improper fractions to mixed fractions and vice-versa. She was curious to know if she could add and subtract improper fractions and mixed fractions too. How do you think Pooja can add or subtract mixed fractions? Recall We have learnt about the types of fractions. Let us recall them here. 1) A fraction whose numerator is greater than the denominator is called an improper fraction. 2) A fraction whose denominator is greater than the numerator is called a proper fraction. 3) The combination of a whole number and a fraction is called a mixed fraction. 26

Let us revise the concept of fractions by solving the following: 13 8 11 5 22 17 a) 6 + 9 b) 7 + 14 c) 15 + 10 8 10 9 23 54 d) 3 – 11 e) 2 – 15 f) 6 – 5 & Remembering and Understanding A mixed fraction can be converted into an improper fraction by multiplying the whole number part by the fraction’s denominator and then adding the product to the numerator. Then we write the result on top of the denominator. The addition and subtraction of mixed fractions are similar to that of unlike fractions. Let us understand the same through the following examples. Example 1: 3 + 3 2 Add: 2 5 7 Solved Solve this Steps 23 + 32 12 1 + 15 1 57 43 Step 1: Convert all the mixed 2 3 = 2 × 5 + 3 = 13 ; fractions into improper fractions. 55 5 3 2 = 3 ×7 + 2 = 23 77 7 Step 2: Find the L.C.M. and add the 2 3 + 3 2 = 13 + 23 improper fractions. 5 7 5 7 [L.C.M. of 5 and 7 is 35.] = 7 ×13 + 5 × 23 35 = 91+115 = 206 35 35 Fractions - II 27

Solved Solve this 12 1 + 15 1 Steps 23 + 32 57 43 Step 3: Find the H.C.F. of the The H.C.F. of 206 and 35 is 1. Solve this numerator and the denominator of So, we cannot reduce the 12 1 from 15 1 the sum. Then reduce the improper fraction any further. fraction to its simplest form. 43 Step 4: Convert the improper fraction 206 31 into a mixed fraction. =5 35 35 Therefore, 2 3 + 3 2 57 = 5 31 . 35 Example 2: Subtract 2 3 from 3 2 57 Steps Solved 2 3 from 3 2 Step 1: Convert all the mixed fractions into improper fractions. 57 3 2 = 3 ×7 + 2 = 23 ; 77 7 2 3 = 2 × 5 + 3 = 13 55 5 Step 2: Find the L.C.M. and 32 -23 = 23 13 subtract the improper fractions. 7 5 7 -5 [L.C.M. of 5 and 7 is 35] = 5 × 23 − 7 ×13 = 115 − 91 = 24 35 35 35 Step 3: Find the H.C.F. of the The H.C.F. of 24 and 35 is 1. So, we numerator and the denominator cannot reduce the fraction any of the difference. Then reduce further. the proper fraction to its simplest form. 28

Steps Solved Solve this 2 3 from 3 2 12 1 from 15 1 57 43 Step 4: If the difference is an 24 is a proper fraction. So, we improper fraction, convert it into 35 a mixed fraction. cannot convert it into a mixed fraction. 2 3 24 Therefore, 3 7 – 2 5 = 35 Fractions - II 29

Concept 10.2: Multiply Fractions Think Pooja and each of her 15 friends had a bar of chocolate. Each of them ate 5 of the 12 chocolate. How much of the chocolate did they eat in all? How do you think Pooja can find this? Recall Recall that when we find the fraction of a number, we multiply the number by the fraction. After multiplication, we simplify the product to its lowest terms. Similarly, we can multiply a fraction by another fraction too. • F raction in its lowest terms: A fraction is said to be in its lowest form if its numerator and denominator do not have a common factor other than 1. • R educing or simplifying fractions: Writing fractions such that its numerator and denominator have no common factor other than 1 is called reducing or simplifying the fraction to its lowest terms. • Methods used to reduce a fraction: A fraction can be reduced to its lowest terms using 1) division 2) H.C. F. Let us revise the concept by simplifying the following fractions. a) 12 b) 16 c) 13 27 24 65 d) 17 e) 9 f) 14 23 21 42 & Remembering and Understanding Multiply fractions by whole numbers A whole number can be considered as a fraction with its denominator as 1. Multiplying a fraction by 2-digit or 3-digit numbers is the same as finding the fraction of a number. Fractions - II 31

Example 7: Find the following: a) 23 of 90 45 b) 15 of 128 32 Solution: a) 23 of 90 = 23 × 90 = 23 × 90 45 45 45 = 2070 = 46 45 Multiplying the numbers in the numerator and then dividing is tedious. It is especially so when the numbers are large. Therefore, we shall find if any of the numbers in the numerator and the denominator have a common factor. If yes, we take the H.C.F. of the numbers. We then divide the numbers to reduce the fraction to its lowest terms. Hence, 23 of 90 = 23 × 90. Here, 45 and 90 have common factors, 3, 5, 9, 15 45 45 and 45. The H.C.F. of 45 and 90 is 45. So, divide both 45 and 90 by their H.C.F. Therefore, 23 × 90 = 23 × 90 2 [Cancelling using the H.C.F. of the numbers] 45 45 1 = 23 × 2 = 46 b) 15 of 128 = 15 × 128 32 32 The H.C.F of 32 and 128 is 32. Divide 32 and 128 by 32, and simplify the multiplication. 15 × 128 4 = 15 × 4 = 60 32 1 Multiply fractions by fractions Multiplication of two fractions is simple. If a and c are two fractions where b,d  are not equal to zero, b d then a × c = a × c b d b × d Product of numerators Therefore, product of the fractions = Product of denominators 32

To multiply mixed number, we change them into improper fractions and then proceed. Multiplying the numbers in the numerator and then dividing is tedious. It is especially so when the numbers are large. Therefore, we shall check if any of the numbers in the numerator and the denominator have a common factor. We then reduce the fractions into their lowest terms and then multiply them. Let us look at an example to understand the concept. Example 8: Solve: 23 × 15 45 46 Solution: Follow these steps to multiply the two fractions. Step 1: Check if the numerator and denominator have any common factors. Observing the given fractions, we see that, a) (23, 45) and (15, 46) do not have any common factors to be reduced. b) (23, 46) and (15, 45) have common factors. Step 2: Find the H.C.F. of the numerator and the denominator that have common factors. The H.C.F. of 23 and 46 is 23. The H.C.F. of 15 and 45 is 15. Step 3: Reduce the numerator and the denominator that have common factors using their H.C.F. 1 23 × 1 = 1×1 =1 15 3 45 46 2 3 × 2 6 Therefore, 23 × 15 = 1 . Example 9: 45 46 6 Solve: a) 2 × 5 b) 7 × 70 c) 84 × 45 56 35 63 54 60 Solution: a) 1 2 × 1 = 1× 1 = 1×1 = 1 15 1 3 1× 3 3 5 63 b) 17 × 2 = 1 × 2 = 1× 2 = 2 70 1 35 63 9 1 9 1× 9 9 c) 7 84 × 5 = 7 × 5 = 7×5 = 7 1 54 6 5 6×5 6 16 6 45 60 5 Fractions - II 33

Concept 11.3: Add and Subtract Decimals Think Pooja went to an ice cream parlour to purchase some ice creams. She bought strawberry for ` 25.50, vanilla for ` 15.30 and chocolate for ` 32.20. She gave ` 100 to the shopkeeper. She wanted to calculate the total price before the shopkeeper gives the bill. Since the prices are in decimals, she was unable to calculate. Do you know how to find the total cost of the ice creams that Pooja bought? How much change would she get in return? Recall Addition and subtraction of decimal numbers is similar to that of usual numbers. Let us recall the conversion of unlike decimals to like decimals. Convert the given unlike decimals into like decimals. a) 4.32, 4.031, 4.1, 7.823 b) 0.7, 0.82, 4.513, 0.72 c) 1.82, 7.01, 5.321, 0.8 d) 7.32, 7.310, 7.8, 5.2 & Remembering and Understanding Add and subtract decimal numbers with the thousandths place Addition and subtraction of decimal numbers with the thousandths place is similar to that of decimals with the hundredths place. Before adding or subtracting any decimals, convert the unlike decimals to like decimal. Write the given decimal numbers such that the digits in their same places are exactly one below the other. Note: The decimal points of the numbers must be exactly one below the other. Decimals - I 55

Let us see a few examples. Example 18: a) Find the sum of 173.809 and 23.617. Solution: b) Subtract 216.735 from 563.726. b) 12 16 a) 5 2/ 6/ 12 11 5 6/ 3/ . 7/ 2/ 6 17 3 . 8 0 9 –2 1 6 . 7 3 5 + 23 . 617 346 . 991 197 . 426 Example 19: Solve: a) 294.631 + 306.524 b) 11.904 – 6.207 Solution: a) 1 1 1 b) 11 8 9 14 294 . 631 1/ 1/ . 9/ 0/ 4/ +3 0 6 . 5 2 4 – 6 . 20 7 601 . 155 5 . 69 7 56

Chapter Decimals - II 12 Let Us Learn About • multiplying and dividing decimals by 1-digit and 2-digit numbers. • m ultiplying decimals by 10, 100 and 1000. • multiplying and dividing a decimal number by another decimal number. • the relationship between percentages, decimals and fractions. Concept 12.1: Multiply and Divide Decimals Think Pooja bought six different types of toys for ` 236.95 each. She calculated the total cost and paid the amount to the shopkeeper. Pooja then went to a sweet shop where 410.750 kg of a sweet was prepared. She wanted to know the number of 250 g packs that can be made from it. Do you know how to find the total cost of the toys? Can you calculate how many packs of sweets can be made? Recall Multiplication and division of decimal numbers are similar to that of usual numbers. Let us recall multiplication and division of numbers by answering the following. Solve: a) 267 × 14 b) 3218 × 34 c) 7424 × 14 d) 576 ÷ 12 e) 265 ÷ 5 f) 384 ÷ 4 61

& Remembering and Understanding Multiplication of decimals is similar to multiplication of numbers. When two decimal numbers are multiplied, a) count the total number of digits after decimal point in both the numbers. Say it is ‘n’. b) multiply the two decimal numbers as usual and place the decimal point in the product after ‘n’ digits from the right. Multiply decimals by 1-digit and 2-digit numbers Let us understand the multiplication of decimals through a few examples. Example 1: Solve: a) 25.146 × 23 b) 276.32 × 6 Solution: a) 25.146 × 23 T Th Th H TO 1 1 1 46 11 23 25 1 38 20 × 58 1 + 75 4 502 9 5 7 8.3 Therefore, 25.146 × 23 = 578.358 b) 276.32 × 6 Step 1: To multiply the given numbers, follow the steps outlined here. Multiply the numbers without considering the decimal point. T Th Th H T O 4 3 11 2 7 6 32 ×6 1 6 5 7 92 62

Step 2:  Count the number of decimal places in the given number. The number of decimal places in 276.32 is two. Step 3: Count from the right, the number of digits in the product as the number of decimal places in the given number. Then place the decimal point. Therefore, 276.32 × 6 = 1657.92. Multiply decimals by 10,100 and 1000 Example 2: Solve: a) 3.4567 × 10 b) 3.4567 × 100 c) 3.4567 × 1000 Solution: To multiply a decimal number by 10, 100 and 1000, follow the steps. Step 1: Write the decimal number as it is. Step 2: Shift the decimal point to the right by as many digits as the number of zeros in the multiplier. Therefore, a) 3 .4567 × 10 = 34.567 (The decimal point is shifted to the right by one digit as the multiplier is 10.) b) 3.4567 × 100 = 345.67 (The decimal point is shifted to the right by two digits as the multiplier is 100.) c) 3 .4567 × 1000 = 3456.7 (The decimal point is shifted to the right by three digits as the multiplier is 1000.) Multiply a decimal number by another decimal number Multiplication of a decimal number by another decimal number is similar to multiplication of a decimal number by a number. Let us understand this through an example. Example 3: Solve: 7.12 × 3.7 Solution: Step1: Multiply the numbers without considering the decimal point. 1 7 12 × 37 11 4 9 84 +2 1 3 6 0 26 3 44 Decimals - II 63

Step 2: Count the number of decimal places in both the multiplicand and the multiplier and add them. The number of decimal places in 7.12 is two The number of decimal places in 3.7 is three Total number of decimal places = 2 + 1 = 3 Step 3: Count as many digits in the product from the right as the total number of decimal places. Then place the decimal point. Therefore, 7.12 × 3.7 = 26.344. Divide decimal numbers by 1-digit and 2-digit numbers Division of decimal numbers is similar to the division of usual numbers. Let us understand this through a few examples. Example 4: Divide: a) 147.9 ÷ 3 b) 64.2 ÷ 6 Solution: Step 1: Follow the steps to divide. Divide the decimal number (dividend) by the 1-digit number (divisor) as usual. Step 2: Place the decimal point in the quotient exactly above the decimal point in Example 5: t he dividend. a) 49.3 b) 10.7 )3 147.9 )6 64.2 −12 −6 27 042 − 27 − 42 09 00 − 09 b) 56.96 ÷ 32 00 Divide: a) 20.475 ÷ 25 Solution : a) 0.819 b) 1.78 )25 20.475 )32 56.96 − 200 − 32 47 249 − 25 − 224 225 256 − 225 − 256 000 000 64

Divide decimals by 10,100 and 1000 Example 6: Solve: a) 3.4567 ÷ 10 b) 3.4567 ÷ 100 c) 3.4567 ÷ 1000 Solution: To divide a decimal number by 10, 100 and 1000, follow these steps: Step 1: Write the decimal number as it is. Step 2: Shift the decimal point to the left by as many digits as the number of zeros in the divisor. Therefore, a) 3 .4567 ÷ 10 = 0.34567 (The decimal point is shifted to the left by one digit as the divisor is 10.) b) 3.4567 ÷ 100 = 0.034567 (The decimal point is shifted to the left by two digits as the divisor is 100.) c) 3.4567 ÷ 1000 = 0.0034567 (The decimal point is shifted to the left by three digits as the divisor is 1000.) Use decimals to continue division of numbers resulting in remainders Recall that sometimes we get remainders in the division of numbers. We can use the decimal point to divide the remainder up to the desired number of decimal places. Let us understand this through a few examples. Example 7: Solve: 54487 ÷ 46 Solution: To divide the given numbers, follow the steps given here. Step 1: Divide as usual, till you get a remainder. 1184 )46 54487 − 46 84 − 46 388 − 368 207 − 184 23 Step 2: Place a point to the right of the quotient. Add a zero to the right of the remainder and continue the division. Decimals - II 65

1184.5 )46 54487 − 46 84 − 46 388 − 368 207 − 184 230 − 230 000 In this case, the division is stopped after one decimal place as the remainder is zero. In some cases, the division continues for more than three decimal places. But usually, we divide up to three decimal places. We then round off the quotient to two decimal places. Example 8: Divide the following up to two decimal places. a) 91158 ÷ 28 b) 78323 ÷ 15 Solution: a) 3255.642 b) 5221.533 )28 91158 )15 78323 − 84 − 75 71 33 − 56 − 30 155 32 − 140 − 30 158 23 − 140 − 15 180 80 − 168 − 75 120 50 − 112 − 45 80 50 − 56 − 45 4 5 Therefore, 91158 ÷ 28 = 3255.64 and 78323 ÷ 15 = 5221.53 after rounding off to two decimal places. 66

Divide a decimal number by another decimal number Let us understand the division of a decimal number by another through an example. Example 9: Solve: 3.0525 ÷ 5.5 Solution: To divide a decimal number by another, follow these steps. Step 1: Convert the decimals into fractions. Step 2: Step 3: 3.0525 = 30525 and 5.5 = 55 Step 4: 10000 10 Find the reciprocal of the divisor. 55 10 Reciprocal of is . 10 55 Multiply dividend by the reciprocal of divisor. 30525 10 30525 555 × = = 10000 55 55´1000 1000 Convert the fraction to a decimal number. 555 = 0.555 1000 Therefore, 3.0525 ÷ 5.5 = 0.555 Decimals - II 67