MAGNOLIA-STUDENT-TEXTBOOK-MATHEMATICS-G04-Combine

MATHEMATICS TEXTBOOK 4 Name: ___________________________________ Section: ________________ Roll No.: _________ School: __________________________________

Contents 4Class Part 1 4 Addition and Subtraction 4.1 A dd and Subtract 5-digit Numbers .......................................................................... 36 5 Multiplication 5.1 M ultiply 3-digit and 4-digit Numbers ....................................................................... 42 7 Division 7.1 Divide Large Numbers............................................................................................... 68

Chapter Addition and Subtraction 4 Let Us Learn About • a dding and subtracting 5-digit numbers. • a pplying addition and subtraction operations in real-life situations. Concept 4.1: Add and Subtract 5-digit Numbers Think In Jasleen’s town, there were 27023 adults and 1567 children. 1400 adults and 1200 children went out of the town on 23rd March 2015. What was the total population of the town on 23rd March? What was the population on the 22nd, if all of them were present in the town that day? Can you also solve it? Recall We know the addition and subtraction of 4-digit numbers. Let us recall the steps followed. Step 1: A rrange the numbers one below the other according to their places. For subtraction, ensure that the smaller number is placed below the larger number. Step 2: Start adding or subtracting from the ones place. 36

Step 3: At every stage, see if regrouping is required and then add or subtract. Step 4: Write the answer. Solve the following to revise the concept. a) Th H T O b) Th H T O c) Th H T O 4216 1335 5985 +1 2 5 9 +1 2 3 5 +2 4 5 3 d) Th H T O e) Th H T O f) Th H T O 7452 4322 6200 –1 3 2 3 –1 4 7 2 –4 5 0 0 & Remembering and Understanding Addition or subtraction of large numbers is similar to the addition or subtraction of 4-digit numbers. We always begin addition and subtraction from the ones place. Let us see an example of addition involving 5-digit numbers. Example 1: Add: 48415 + 20098 Solution: Arrange the numbers one below the other. Steps Solved Solve these T Th Th H T O Step 1: Add the tens and ones. T Th Th H T O Write the sum under the ones. Regroup if needed. 1 4 8415 5 7383 +3 1347 +2 0098 3 Addition and Subtraction 37

Steps Solved Solve these Step 2: Add the tens and also T Th Th H T O T Th Th H T O the carry forward (if any) from 11 the previous step. Write the 2 5347 sum under the tens. Regroup if 4 8415 +6 2567 needed. +2 0098 513 Step 3: Add the hundreds T Th Th H T O T Th Th H T O and also the carry forward (if any) from the previous 11 step. Write the sum under the hundreds. Regroup if 4 8415 4 2688 needed. +1 2912 +2 0098 513 Step 4: Add the thousands T Th Th H T O T Th Th H T O and also the carry forward 11 (if any) from the previous 3 4765 step. Write the sum under 4 8415 +2 1178 the thousands. Regroup if +2 0098 needed. 8513 Step 5: Add the ten thousands T Th Th H T O T Th Th H T O and also the carry forward (if any) from the previous step. 11 Write the sum under the ten thousands. 4 8415 8 2633 Thus, 48415 + 20098 = 68513. +1 0120 +2 0098 6 8513 We will now learn subtraction of 5-digit numbers. Example 2: Subtract: 56718 – 16754 Solution: Arrange the numbers in columns. 38

Steps Solved Solve these T Th Th H T O T Th Th H T O Step 1: Subtract the ones and write the difference under the 5 6718 9 7054 ones. −1 6754 – 2 3567 4 T Th Th H T O 7 5400 Step 2: Subtract the tens. That is, T Th Th H T O 1 − 5, which is not possible. – 3 2689 5 6⁄ 1⁄1 Regroup the hundreds to −1 T Th Th H T O tens, subtract and write the 6718 8 5464 difference under the tens. 6754 – 1 2078 64 T Th Th H T O Step 3: Subtract the hundreds. T Th Th H T O 5 4635 That is, 6 − 7, which is not possible. 1⁄6 – 1 2789 5⁄ 6⁄ 1⁄1 Regroup the thousands to T Th Th H T O hundreds, subtract and write the 5 6718 8 9576 difference under the hundreds. −1 6754 – 4 5689 964 Step 4: Subtract the thousands. T Th Th H T O That is, 5 − 6, which is not possible. 4 165⁄⁄5 176⁄⁄6 11⁄1 8 5 Regroup the ten thousands to thousands, subtract and −1 6754 write the difference under the thousands. 9964 Step 5: Subtract the ten T Th Th H T O thousands, and write the difference under the ten 54⁄ 165⁄⁄5 167⁄⁄6 11⁄1 8 thousands −1 6754 Thus, 56718 – 16754 = 39964. 3 9964 Addition and Subtraction 39

Chapter Multiplication 5 Let Us Learn About • multiplying 3-digit and 4-digit numbers. • p roperties of multiplication. • m ultiplying using standard and lattice algorithms. • multiplying mentally. Concept 5.1: Multiply 3-digit and 4-digit Numbers Think Jasleen went to the stadium to watch a rugby match with her parents. She observed that the seats are arranged in many rows and columns. All the seats were occupied. She wanted to guess the total number of people who watched the match that day. How will she be able to do that? Recall We have learnt to multiply 2-digit and 3-digit numbers by 1-digit and 2-digit numbers. 42

Let us solve the following to revise the concept of multiplication. TO H TO H TO H TO 39 256 589 875 ×2 ×3 ×4 ×5 & Remembering and Understanding Standard algorithm is the method of multiplication in which the product is regrouped as ones and tens. Let us now learn to multiply 3-digit numbers by 3-digit numbers and 4-digit numbers by 1-digit numbers using standard algorithm. Multiply a 3-digit number by a 3-digit number Multiplying a 3-digit number by a 3-digit number is similar to multiplying a 3-digit number by a 2-digit number. Let us see an example. Example 1: Multiply: 159 × 342 Solution: To multiply the given numbers, follow these steps. Steps Solved Solve these Step 1: Multiply the multiplicand by the ones of the Th H T O T Th Th H T O multiplier, that is, 159 × 2. 526 Regroup if necessary. 11 159 ×235 Step 2: Put a 0 below the ones ×342 place of the product obtained 318 in the above step. Multiply the multiplicand by the tens of the Th H T O multiplier, that is, 159 × 4. Regroup if necessary. 23 11 159 ×342 318 6360 Multiplication 43

Steps Solved Solve these Step 3: Put two 0s below the T Th Th H T O ones and the tens places of T Th Th H T O the product obtained in the 425 previous step. Multiply the 12 ×119 multiplicand by the hundreds of the multiplier, that is, 159 × 3. 23 T Th Th H T O Regroup if necessary. 301 11 Step 4: Add the products from 159 ×769 steps 1, 2 and 3. This sum gives ×342 the required product. 318 6360 4 7700 T Th Th H T O 12 23 11 159 ×342 11 3 1 8 + 6360 +4 7 7 0 0 54 378 Multiply a 4-digit number by a 1-digit number Multiplying a 4-digit number by a 1-digit number is similar to multiplying a 3-digit number by a 1-digit number. Let us see an example. Example 2: Multiply: 3628 × 7 Solution: T Th Th H T O 4 15 3 628 ×7 2 5 396 44

Th H T O Solve these Th H T O Th H T O 2568 1259 ×8 5689 ×4 ×2 Properties of Multiplication Identity Property: For any number ‘a’, a × 1 = 1 × a = a. 1 is called the multiplicative identity. For example, 461 × 1 = 1 × 461 = 461. Zero Property: For any number ‘a’, a × 0 = 0 × a = 0. For example, 568 × 0 = 0 × 568 = 0. Commutative Property: If ‘a’ and ‘b’ are any two numbers, then a × b = b × a. For example, 12 × 3 = 36 = 3 × 12. 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 (3 × 4) × 5 = 3 × (4 × 5) Distributive Property: 1) If 'a', 'b' and 'c' are any three numbers, then: a × (b + c) = (a × b) + (a × c). For example, 2 × (3 + 5) = (2 × 3) + (2 × 5). 2 × 8 = 6 + 10 16 = 16 Multiplication distributes over addition. Multiplication 45

2) If 'a', 'b' and 'c' are any three numbers then: a × (b − c) = (a × b) − (a × c). For example, 2 × (8 − 5) = (2 × 8) − (2 × 5). 2 × 3 = 16 − 10 6=6 Multiplication distributes over subtraction. 46

Chapter Division 7 Let Us Learn About • dividing 4-digit numbers by 1-digit and 2-digit numbers. • dividing 3-digit numbers by 2-digit numbers. • properties of division. Concept 7.1: Divide Large Numbers Think Jasleen and seven of her friends want to share 3540 papers equally among themselves. Do you think the papers can be divided, without some being left over? Recall Recall that we can write two multiplication facts for a division fact. For example, a multiplication fact for 45 ÷ 9 = 5 can be written as 9 × 5 = 45 or 5 × 9 = 45. 45 ÷ 9 = 5 ↓ ↓ ↓ Dividend Divisor Quotient The number that is divided is called the dividend. The number that divides is called the divisor. The number of times the divisor divides the dividend is called the quotient. 68

Factors Factors Multiplicand × Multiplier = Product Multiplicand × Multiplier = Product 5 × 9 = 45 9 × 5 = 45 ↓ ↓ ↓ ↓ ↓ ↓ Divisor Quotient Dividend Divisor Quotient Dividend The part of the dividend that remains without being divided is called the remainder. Let us solve the following to revise the concept of division. a) 72 ÷ 9 b) 42 ÷ 3 c) 120 ÷ 5 d) 80 ÷ 4 e) 24 ÷ 1 & Remembering and Understanding In Class 3, we have learnt that division and multiplication are reverse operations. Let us now understand the division of large numbers using multiplication. Division of a 4-digit number by a 1-digit number Dividing a 4-digit number by a 1-digit number is similar to that of a 3-digit number by a 1-digit number. Example 1: Solve: 2065 ÷ 5 Solution: Steps Solved Solve these Step 1: Check if the thousands digit of the dividend is greater than the divisor. If it is )5 2065 )7 3748 not, consider the hundreds digit also. 2 is not greater than Dividend = _____ Step 2: Find the largest number in the 5. So, consider 20. Divisor = ______ multiplication table of the divisor that can Quotient = ____ be subtracted from the 2-digit number of 4 Remainder = ___ the dividend. Write the quotient. Write the product of the quotient and divisor below )5 2065 the dividend. -2 0 Step 3: Subtract and write the difference. 5 × 4 = 20 5 × 5 = 25 25 > 20 4 )5 2065 -20 0 Division 69

Steps Solved Solve these Step 4: Check if difference < 0 < 5 (True) divisor is true. )3 2163 4 If it is false, the division is incorrect. Dividend = _____ Step 5: Bring down the tens digit of the )5 2065 Divisor = ______ dividend and write it near the remainder. Quotient = ____ −20↓ Remainder = ___ 06 )5 1555 Step 6: Find the largest number in the 5×1=5 multiplication table of the divisor that can 5 × 2 = 10 Dividend = _____ be subtracted from the 2-digit number in 5 < 6 < 10 Divisor = ______ the previous step. So, 5 is the required Quotient = ____ number. Remainder = ___ Step 7: Write the factor of the required 41 number, other than the divisor, as the quotient. )5 2 0 6 5 Write the product of the divisor and the − 20 ↓ quotient below the 2-digit number. 06 Then subtract them. − 05 01 Step 8: Repeat steps 6 and 7 till all the digits 1 < 5 (True) of the dividend are brought down. 4 13 Check if remainder < divisor is true. )5 2 0 6 5 Stop the division. (If this is false, the division is incorrect.) −2 0 ↓ 06 − 05 0 15 − 015 000 Step 9: Write the quotient and the Quotient = 413 remainder. Remainder = 0 Step 10: Check if (Divisor × Quotient) + 5 × 413 + 0 = 2065 Remainder = Dividend is true. If this is false, 2065 + 0 = 2065 the division is incorrect. 2065 = 2065 (True) 70

Division of a 3-digit number by a 2-digit number Let us understand the division of 3-digit numbers by 2-digit numbers, through some examples. Example 2: Divide: 414 ÷ 12 Solution: )Write the dividend and the divisor as Divisor Dividend Steps Solved Solve these Step 1: Guess the quotient by thinking of )12 414 dividing 41 by 12. )14 324 Find the multiplication fact which has 12 × 3 = 36 the number less than or equal to the 12 × 4 = 48 dividend and the divisor. 36 < 41 < 48 So, 36 is the number to be subtracted from 41. Step 2: Write the factor other than the Write 3 in the quotient and Dividend = _____ Divisor = ______ dividend and the divisor as the quotient. 36 below 41, and subtract. Quotient = ____ Then bring down the next number in the dividend. 3 )12 414 −36 ↓ 054 Remainder = ___ Step 3: Guess the quotient by thinking of 12 × 4 = 48 )16 548 dividing 54 by 12. 12 × 5 = 60 Dividend = _____ Divisor = ______ Find the multiplication fact which has 48 < 54 < 60 Quotient = ____ the number less than or equal to the So, 48 is the number to be Remainder = ___ dividend and divisor. Write the factor subtracted from 54. other than the dividend and the divisor as the quotient. Write 4 in the quotient and 48 below 54, and subtract. 34 )12 414 −36 ↓ 054 − 048 6 Quotient = 34 Remainder = 6 Division 71

Checking for the correctness of division: We can check whether our division is correct or not using a multiplication fact of the division. Step 1: Compare the remainder and the divisor. [Note: The remainder must always be less than the divisor.] Step 2: Check if (Quotient × Divisor) + Remainder = Dividend Let us now check if our division in example 2 is correct. Steps Checked Step 1: Remainder < Divisor Dividend = 414 Step 2: (Quotient × Divisor) + Divisor = 12 Remainder = Dividend Quotient = 34 Remainder = 6 6 < 12 (True) 34 × 12 + 6 = 414 408 + 6 = 414 414 = 414 (True) Note: a) If remainder > divisor, the division is incorrect. b) If (Quotient × Divisor) + Remainder is not equal to Dividend, the division is incorrect. Dividing a 4-digit number by a 2-digit number Dividing a 4-digit number by a 2-digit number is similar to dividing a 3-digit number by a 2-digit number. Let us understand this through the following example. Example 3: Solve: 2340 ÷ 15 Solution: Steps Solved Solve these Step 1: Check if the thousands digit 2 is not greater than 15. So, )12 5088 of the dividend is greater than the consider 23. divisor. If it is not, consider also the hundreds digit too. )15 2340 72

Steps Solved Solve these Step 2: Guess the quotient by 1 Dividend = _____ thinking of dividing 23 by 15. Divisor = ______ )15 2340 Quotient = _____ Remainder = _____ Find the multiplication fact which has −15 )14 4874 a number less than or equal to the 15 × 1 = 15 dividend and the divisor. 15 × 2 = 30 15 < 23 < 30 So, 15 is the required number. Step 3: Write the factor other than Write 1 in the quotient and 15 the dividend and the divisor as the below 23 and subtract. Then quotient. bring down the next number in the dividend. 1 )15 2340 −15 ↓ 84 Step 4: Guess the quotient by 15 × 5 = 75 thinking of dividing 84 by 15. 15 × 6 = 90 Find the multiplication fact which has 75 < 84 < 90 Dividend = _____ a number less than or equal to the Divisor = ______ So, 75 is the required number Quotient = _____ dividend and the divisor. Remainder = _____ that is to be subtracted from Write the factor other than the dividend and the divisor as the 84. 156 quotient. )15 2340 − 15↓ 84 − 75 9 Division 73

Steps Solved Solve these Step 5: Subtract and write the 15 × 5 = 75 )16 3744 difference. Repeat till all the digits of 15 × 6 = 90 the dividend are brought down. 90 = 90 So, 90 is the required number. 156 )15 2340 − 15↓ Dividend = _____ 84 Divisor = ______ Quotient = _____ − 75 90 − 90 00 Quotient = 156 Remainder = 0 Step 6: Check if (Divisor × Quotient) + 15 × 156 + 0 = 2340 Remainder = _____ Remainder = Dividend is true. If this is 2340 + 0 = 2340 false, the division is incorrect. 2340 = 2340 (True) Let us see some properties of division. Properties of division 1) Dividing a number by 1 gives the same number as the quotient. For example: 15 ÷ 1 = 15; 1257 ÷ 1 = 1257; 1 ÷ 1 = 1; 0 ÷ 1 = 0 2) Dividing a number by itself gives the quotient as 1. For example: 15 ÷ 15 = 1; 1257 ÷ 1257 = 1; 1 ÷ 1 = 1 3) Division by zero is not possible and is not defined. For example: 10 ÷ 0; 1257 ÷ 0; 1 ÷ 0 are not defined 74

Contents 4Class Part 2 8 Fractions - I 8.1 Equivalent Fractions .................................................................................................... 1 8.3 Add and Subtract Like Fractions .............................................................................. 11 10 Decimals 10.1 Conversion Involving Fractions ............................................................................... 28

Chapter Fractions - I 8 Let Us Learn About • e quivalent fractions. • p roblems related to equivalent fractions. • like and unlike fractions. • a dding and subtracting like fractions. Concept 8.1: Equivalent Fractions Think Jasleen cuts 3 apples into 18 equal pieces. Ravi cuts an apple into 6 equal pieces. Did both of them cut the apples into equal pieces? Recall In Class 3, we have learnt that a fraction is a part of a whole. A whole can be a region or a collection. When a whole is divided into two equal parts, each part is called ‘a half’. 11 22 ‘Half’ means 1 out of 2 equal parts. We write ‘half’ as 1 . 2 1

Two halves make a whole. Numerator Numbers of the form Denominator are called fractions. The total number of equal parts into which a whole is divided is called the denominator. The number of such equal parts taken is called the numerator. Similarly, each of the three equal parts of a whole is called a third. We write one-third as 1 and, two-thirds as 2 . 33 Three-thirds or 3 make a whole. 3 Each of four equal parts of a whole is called a fourth or a quarter written as 1 . 4 Two such equal parts are called two-fourths, and three equal parts are called three-fourths, written as 2 and 3 respectively. Four quarters make a whole. 44 2 halves, 3 thirds, 4 fourths, 5 fifths, …, 10 tenths make a whole. So, we write a whole as 2 , 3 , 4 , 5 ,...,10 and so on. 2 3 4 5 10 & Remembering and Understanding Fractions that denote the same part of a whole are called equivalent fractions. Let us now understand what equivalent fractions are. Suppose there is 1 bar of chocolate with Ram and Raj each as shown. chocolate with Ram chocolate with Raj Ram eats 1 of the chocolate. 5 Then the piece of chocolate he gets is Raj eats 2 of the chocolate. 10 Then the piece of chocolate he gets is 2

We see that both the pieces of chocolates are of the same size. So, we say that the fractions 1 and 2 are equivalent. We write them as 1 = 2 . 5 10 5 10 Example 1: Shade the regions to show equivalent fractions. a) [ 1 and 2 ] 36 b) 1 2] [ and 48 Solution: a) 1 3 2 6 b) 1 4 2 8 Example 2: Find the figures that represent equivalent fractions. Also, mention the fractions. a) b) c) d) Fractions - I 3

Solution: The fraction represented by the shaded part of figure a) is 1 . 2 The shaded part of figure b) represents 2 . The shaded part of figure d) 4 represents 1 . 2 So, the shaded parts of figures a), b) and d) represent equivalent fractions. 4

Concept 8.3: Add and Subtract Like Fractions Think Jasleen has a cardboard piece, equal parts of which are coloured in different colours. Some of the equal parts are not coloured. She wants to find the part of the cardboard that has been coloured and left uncoloured. How do you think Jasleen can find that? Recall Recall that like fractions have the same denominators. To compare them, we compare their numerators. Let us answer the following to recall the concept of like fractions. Compare the following using >, < and =. a) 2 ____ 1 b) 4 ____ 8 c) 3 ____ 5 d) 7 ____ 3 e) 1 ____ 4 33 10 10 77 88 55 & Remembering and Understanding While adding or subtracting like fractions, add or subtract only their numerators. Write the sum or difference on the same denominator. Let us understand addition and subtraction of like fractions through some examples. Example 13: In the given figures, find the fractions represented by the shaded parts using addition. Then find the fractions represented by the unshaded parts using subtraction. a) b) c) Solution: We can find the fractions represented by the shaded and the unshaded parts with the following steps. Fractions - I 11

Solved Solve these Steps Step 1: Count the total Total number of equal Total number of Total number of number of equal parts. equal parts = ____ parts = 6 equal parts = ___ Step 2: Count the a) Number of parts a) Number of parts a) N umber of parts number of parts of each coloured pink = 1 coloured yellow coloured violet = colour. = ______ _______ b) Number of parts coloured blue = 2 b) Number of parts b) N umber of parts coloured violet = coloured brown _______ = ______ Step 3: Write the fraction Pink: 1 , Blue: 2 Yellow: ________ Violet: ________ representing the number 66 Violet: ________ Brown: ________ of parts of each colour. Step 4: To add the like The fraction that The fraction that The fraction that fractions in step 3, add represents the their numerators and represents the shaded represents the shaded part of the write the sum on the given figure is same denominator. part of the given shaded part of the ____ + ____=____. figure is given figure is 1 + 2 = 1+ 2 = 3 . ____ + ____=____. 66 6 6 Step 5: Write the whole Like fraction Like fraction Like fraction representing the representing the as a like fraction of the representing the whole = 6 . whole = _______. whole = _______. sum in step 4. Then, to 6 So, the fraction So, the fraction So, the subtract the like fractions, that represents the that represents the subtract their numerators. unshaded part of the unshaded part of fraction that Write the difference on given figure is the given figure is represents the the same denominator. unshaded part of ____ − ____=____. the given figure is 6−3 =6−3 = 3. 66 6 6 ____ − ____=_____. 12

Example 14: Add: a) 3 + 1 45 23 57 88 b) + c) + Solution: a) 3 + 1 = 3 + 1 = 4 13 13 100 100 88 8 8 c) 48 – 26 b) 4 + 5 = 4 + 5 = 9 125 125 13 13 13 13 c) 23 + 57 = 23 + 57 = 80 100 100 100 100 Example 15: Subtract: a) 8 – 4 b) 33 – 25 99 37 37 Solution: a) 8 – 4 = 4 99 9 b) 33 – 25 = 33 − 25 = 8 37 37 37 37 48 26 48 − 26 22 c) – = = 125 125 125 125 Fractions - I 13

Chapter Decimals 10 Let Us Learn About • the term ‘decimal’ and its parts. • u nderstanding decimal system. • expanding decimal numbers with place value charts. • converting fractions to decimals and vice versa. Concept 10.1: Conversion Involving Fractions Think Jasleen and her friends participated in the long jump event in their Jasleen – 4.1m Ravi – 2.85m games period. Her sports teacher noted the distance they jumped on a Rajiv – 3.05 m piece of paper as shown here. Amit – 2.50m Jasleen wondered why the numbers had a point between them as in the case of writing money. Do you know what the point means? Recall Recall that in Class 3 we have learnt to measure the lengths, weights and volumes of objects. For example, a pencil is 12.5 cm long. 12. 5 cm 28

A crayon is 5.4 cm long. 5.4 cm The weight of your mathematics textbook is 0.905 kg. A milk packet has 0.250 of milk, and so on. In all these values, we see numbers with a point between them. Have you read price tags on some items when you go shopping? ` 300.75 ` 439.08 They also have numbers with a point between them. Let us learn why a point is used in such numbers. & Remembering and Understanding We know how to write fractions. In this figure, 3 portion is coloured and 7 portion is not coloured. 10 10 3 or 0.3 and the We can write the coloured portion of the figure as 10 portion that is not coloured as 7 or 0.7. 10 Numbers such as 0.3, 0.7, 3.0, 3.1, 4.7, 58.2 and so on are called decimal numbers or simply decimals. Tenths: The figure below is divided into ten equal parts. 1 111 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 Each column is of the same size. Then, each of the ten equal parts is 1 . It is read as one-tenth. Fractional form of each equal part is 1 . 10 10 Decimal form of each equal part is 0.1. Decimals 29

We read numbers like 0.1, 0.2, 0.3 … as ‘zero point one’, ‘zero point two’, ‘zero point three’ and so on. Zero is written to indicate the place of the whole number. A decimal number has two parts. 48 . 35 Whole or integral part Decimal part (= or > 0) (< 1) Decimal Point Note: T he numbers in the decimal part are read as separate digits. Recall the place value chart of numbers. 100 × 10 10 × 10 1 × 10 1 Thousands Hundreds Tens Ones 6 2 5 5 3 2 2 6 5 2 We know that in this chart, as we move from right to left, the value of the digit increases 10 1 times. Also, as we move from left to right, the value of a digit becomes times. The place 10 value of the digit becomes one-tenth, read as a tenth. Its value is 0.1 read as ‘zero point one’. 2 is read as ‘two-tenths’, 7 is read as ‘seven–tenths’ and so on. 10 10 We can extend the place value chart to the right as follows: 1 × 1000 1 × 100 1 × 10 1 . 1 10 Thousands Hundreds Tens Ones Decimal Tenths 7 . 2 1 2 4 . 3 30 4 3 . 6 1 5 . 7 The number 3015.7 is read as three thousand and fifteen point seven. Similarly, the other numbers are read as follows: 30

Seven point two; twenty-four point three and one hundred and forty-three point six. The point placed in between the number is called the decimal point. The system of writing numbers using a decimal point is called the decimal system. [Note: ‘Deci’ means 10.] Hundredths: Study this place value chart. Thousands Hundreds Tens Ones Decimal Tenths Hundredths 1 × 10 1 point 1 × 1000 1 × 100 2 1 1 2 8 6 . 10 100 3 . 9 When the number moves right from the tenths place, we get a new place, which is 1 of the tenths place. It is called the ‘hundredths’ place written as 1 and read 10 100 as one-hundredths. Its value is 0.01, read as ‘zero point zero one’. 2 is read as two-hundredths, 5 is read as five-hundredths and so on. 100 100 So, the number in the place value chart is read as ‘two thousand eight hundred and sixty-two point three nine’. Expansion of decimal numbers Using the place value chart, we can expand decimal numbers. Let us see a few examples. Example 1: Expand these decimals. a) 1430.8 b) 359.65 c) 90045.75 d) 654.08 Solution: To expand the given decimal numbers, first write them in the place value chart as shown. S. no Ten Thousands Hundreds Tens Ones Decimal Tenths Hundredths thousands 1 point a) 4 3 0 . 8 b) 9 0 3 5 9 65 c) 0 4 5 . 75 d) 6 5 4 08 . . Decimals 31

Expansions: 1 a) 1430.8 = 1 × 1000 + 4 × 100 + 3 × 10 + 0 × 1 + 8 × 10 Example 2: b) 359.65 = 3 × 100 + 5 × 10 + 9 × 1 + 6 × 1 + 5 × 1 10 100 c) 90045.75 = 9 × 10000 + 0 × 1000 + 0 × 100 + 4 × 10 + 5 × 1 + 7 × 1 + 5 × 1 10 100 1 1 Solution: d) 654.08 = 6 × 100 + 5 × 10 + 4 × 1 + 0 × + 8 × 10 100 Write these as decimals. a) 7 × 1000 + 2 × 100 + 6 × 10 + 3 × 1 + 9 × 1 + 3 × 1 10 100 b) 3 × 10000 + 0 × 1000 + 1 × 100 + 9 × 10 + 6 × 1 + 4 × 1 + 5 × 1 10 100 c) 2 × 1000 + 2 × 100 + 2 × 10 + 2 × 1 + 2 × 1 + 2 × 1 10 100 d) 5 × 100 + 0 × 10 + 0 × 1 + 0 × 1 + 5 × 1 10 100 First write the numbers in the place value chart as shown. S. no Ten Thousands Hundreds Tens Ones Decimal Tenths Hundredths thousands point a) 7 2 63 . 93 b) 3 0 1 96 . 45 c) 2 2 22 . 22 d) 5 00 . 05 Standard forms of the given decimals are: a) 7263.93 b) 30196.45 c) 2222.22 d) 500.05 Conversion of fractions to decimals Fractions can be written as decimals. Consider an example. Example 3: Express these fractions as decimals. a) 18 2 b) 43 5 c) 26 1 d) 4 9 10 10 10 10 Solution: To write the given fractions as decimals, follow these steps. Step 1: Write the integral part as it is. Step 2: Place a point to its right. 32

Step 3: Write the numerator of the proper fraction part. a) 18 2 = 18.2 b) 43 5 = 43.5 10 10 c) 26 1 = 26.1 d) 4 9 = 4.9 10 10 Example 4: Express these fractions as decimals. a) 25 b) 17 2 c) 43 d) 5 92 100 100 100 100 Solution: a) 25 = 25 hundredths = 0.25 100 b) 17 2 = 17 and 2 hundredths = 17.02 100 c) 43 = 43 hundredths = 0.43 100 d) 5 92 = 5 and 92 hundredths = 5.92 100 Shortcut method: Fractions having 10 or 100 as their denominators, can be expressed in their decimal form by following the steps given below. Step 1: Write the numerator. Step 2: Then count the number of zeros in the denominator. Step 3: Place the decimal point after the same number of digits from the right as the number of zeros. For example, the decimal form of 232 = 2.32 100 Note: F or the decimal equivalent of a proper fraction, place a 0 as the integral part of the decimal number. Conversion of decimals to fractions To convert a decimal into a fraction, follow these steps. Step 1: Write the number without the decimal. Step 2: Count the number of decimal places (that is, the number of places to the right of the decimal number). Step 3: Write the denominator with 1 followed by as many zeros as the decimal point. Decimals 33

Example 5: Write these decimals as fractions. a) 2.3 b) 13.07 c) 105.43 d) 0.52 Solution: a) 2.3 = 23 b) 13.07 = 1307 10 100 c) Alternate method: 105.43 = 10543 d) 0.52 = 52 100 100 A decimal having an integral part can be written as a mixed fraction. So, 2.3 = 2 and 3 tenths = 2 3 10 13.07 = 13 and 7 hundredths = 13 7 100 105.43 = 105 and 43 hundredths = 105 43 100 34