MATHEMATICS TEXTBOOK 3 Name: ___________________________________ Section: ________________ Roll No.: _________ School: __________________________________

Contents 3Class Part 1 18 24 3 Numbers 35 3.1 Count by Thousands 48 3.2 Compare 4-digit Numbers 59 4 Addition 63 4.2 Add 3-digit and 4-digit Numbers 5 Subtraction 5.2 Subtract 3-digit and 4-digit Numbers 6 Multiplication 6.1 Multiply 2-digit Numbers 6.2 Multiply 3-digit Numbers by 1-digit and 2-digit Numbers

Chapter Numbers 3 Let Us Learn About • writing 4-digit numbers with place value chart. • identifying and forming the greatest and the smallest number. • writing the standard and the expanded forms of the number. • comparing and ordering numbers. Concept 3.1: Count by Thousands Think Farida went to buy one of the toy cars shown. She could not read the price on one of the cars. Can you read the price on ` 1937.00 both the cars and understand what they mean? ` 657.00 Recall We know that 10 ones make a ten. Similarly, 10 tens make a hundred. Let us now count by tens and hundreds as: Counting by 10s: 10, 20, 30, 40, 50, 60, 70, 80 and 90 Counting by 100s: 100, 200, 300, 400, 500, 600, 700, 800 and 900 18

When we multiply a digit by the value of its place, we get its place value. Using place values, we can write a number in its expanded form. Let us answer these to revise the concept. a) The number for two hundred and thirty-four is _____________. b) In 857, there are _______ hundreds, _______ tens and _______ ones. c) The expanded form of 444 is _______________________. d) The place value of 9 in 493 is _____________. e) The number name of 255 is _______________________________________. & Remembering and Understanding To know about 4-digit numbers, we count by thousands using boxes. Suppose shows 1. Ten such boxes show a 10. So, = 10 ones = 1 ten Similarly, 10 such strips show 10 tens or 1 hundred. = 10 tens = 1 hundred Numbers 19

= 1 hundred = 100 = 2 hundreds = 200 = 3 hundreds = 300 = 4 hundreds = 400 In the same way, we get 5 hundreds = 500, 6 hundreds = 600, 7 hundreds = 700, 8 hundreds = 800 and 9 hundreds = 900. Using a spike abacus and beads of different colours, we represent 999 as shown. 9 blue, 9 green and 9 pink beads on the abacus represent 999. H TO Remove all the beads and Th H T O represent 999 put an orange bead on the represent 1000 next spike. This represents one thousand. We write it as 1000. 1000 is the smallest 4-digit number. Now, we know four places: ones, tens, hundreds and thousands. Let us represent 4732 in the place value chart. 20

Thousands (Th) Hundreds (H) Tens (T) Ones (O) 4 7 32 We count by 1000s as 1000 (one thousand), 2000 (two thousand)... till 9000 (nine thousand). The greatest 4-digit number is 9999. Expanded form of 4-digit numbers The form in which a number is written as the sum of the place values of its digits is called its expanded form. Let us now learn to write the expanded form of 4-digit numbers. Example 1: Expand the following numbers. a) 3746 b) 6307 Solution: Write the digits of the given numbers in the place value chart, as shown. Expanded forms: Th H TO a) 3746 = 3000 + 700 + 40 + 6 a) 3 7 46 b) 6307 = 6000 + 300 + 0 + 7 b) 6 3 0 7 Writing number names of 4-digit numbers Observe the expanded form and place value chart for a 4-digit number, 8015. Th H TO Place values 80 15 5 ones = 5 1 tens = 10 0 hundreds = 0 8 thousands = 8000 We can call 8015 as the standard form of the number. Let us look at an example. Example 2: Write the expanded forms and number names of these numbers. a) 1623 b) 3590 Numbers 21

Solution: To expand the given numbers, write them in the correct places in the place value chart. Expanded forms: Th H T O a) 1623 = 1000 + 600 + 20 + 3 a) 1 6 2 3 b) 3590 = 3000 + 500 + 90 + 0 b) 3 5 9 0 Writing in words (Number names): a) 1623 = One thousand six hundred and twenty-three b) 3590 = Three thousand five hundred and ninety We can write the standard form of a number from its expanded form. Let us see an example. Example 3: Write the standard form of 3000 + 400 + 60 + 5. Solution: Write the numbers in the place value Th H T O chart in the correct places. Write the 3 46 5 digits one beside the other, starting from the thousands place. 3000 + 400 + 60 + 5 = 3465 So, the standard form of 3000 + 400 + 60 + 5 is written as 3465. 22

Concept 3.2: Compare 4-digit Numbers Think Farida has 3506 paper clips and her brother has 3605 paper clips. Farida wants to know who has more paper clips. But the numbers appear to be the same, and she is confused. Can you tell who has more number of paper clips? 24

Recall In class 2, we have learnt to compare 3-digit numbers and 2-digit numbers. Let us quickly revise the concept. A 2-digit number is always greater than a 1-digit number. A 3-digit number is always greater than a 2-digit number and a 1-digit number. So, a number with more number of digits is always greater than a number with lesser digits. We use the symbols >, < or = to compare two numbers. & Remembering and Understanding Comparing two 4-digit numbers is similar to comparing two 3-digit numbers. Let us understand the steps to compare through an example. Example 9: Compare: 5382 and 5380 Solution: Follow these steps to compare the given numbers. Steps Solved Solve this Step 1: Compare the number of digits 5382 and 5380 7469 and 7478 Count the number of digits in the given numbers. The number having more number of digits is Both 5382 and greater. 5380 have 4 digits. Step 2: Compare thousands If two numbers have the same number of digits, 5=5 ____ = ____ compare the thousands digits. (If two numbers have an equal number of digits, start comparing 3=3 ____ = ____ from the leftmost digit.) The number with the greater digit in the thousands place is greater. Step 3: Compare hundreds If the digits in the thousands place are the same, compare the digits in the hundreds place. The number with the greater digit in the hundreds place is greater. Numbers 25

Steps Solved Solve this 5382 and 5380 7469 and 7478 Step 4: Compare tens 8=8 ____ > ____ If the digits in the hundreds place are also same, So, compare the digits in the tens place. The number with the greater digit in the tens place is greater. ____ > ____ Step 5: Compare ones If the digits in the tens place are also the same, 2>0 compare the digits in the ones place. The So, - number with the greater digit in the ones place is greater. When the ones place are the same, the 5382 > 5380 numbers are equal. Note: Once we can decide a greater/smaller number, the steps that follow need not be carried out. 26

Concept 4.2: Add 3-digit and 4-digit Numbers Think Farida’s father bought her a shirt for ` 335 and a skirt for ` 806. Farida wants to find how much her father had spent in all. How do you think she can find that? Recall We can add 2-digit or 3-digit numbers by writing them one below the other. This method of addition is called vertical addition. Let us revise the earlier concept and solve the following. a) 22 + 31 = _________ b) 42 + 52 = _________ c) 82 + 11 = _________ d) 101 + 111 = _________ e) 100 + 200 = _________ f) 122 + 132 = _________ Addition 35

& Remembering and Understanding Let us now understand the addition of two 3-digit numbers with regrouping. We will also learn to add two 4-digit numbers. Add 3-digit numbers with regrouping Sometimes, the sum of the digits in a place is more than 9. In such cases, we need to regroup the sum. We then carry forward the digit to the next place. Example 8: Add 245 and 578. Solution: Arrange the numbers one below the other. Regroup if the sum of the digits is more than 9. Step 1: Add the ones. Solved Step 3: Add the hundreds. H TO Step 2: Add the tens. H TO 1 11 245 H TO 245 11 +578 245 +578 3 +578 823 23 H TO Solve these H TO H TO 823 171 +197 390 +219 +121 Add 4-digit numbers without regrouping Adding two 4-digit numbers is similar to adding two 3-digit numbers. Let us understand this through an example. Example 9: Add 1352 and 3603. Solution: Arrange the numbers one below the other. 36

Solved Step 1: Add the ones. Step 2: Add the tens. Th H T O Th H T O 135 2 135 2 +3 6 0 3 +3 6 0 3 5 55 Step 3: Add the hundreds. Step 4: Add the thousands. Th H T O Th H T O 135 2 13 5 2 +3 6 0 3 +3 6 0 3 955 49 5 5 Th H T O Solve these Th H T O 41 9 0 11 1 1 +2 0 0 0 Th H T O +2 2 2 2 200 2 +3 0 0 3 Add 4-digit numbers with regrouping We regroup the sum when it is equal to or more than 10. Example 10: Add 1456 and 1546. Solution: Arrange the numbers one below the other. Add and regroup, if necessary. Solved Step 1: Add the ones. Step 2: Add the tens. Th H T O Th H T O 1 11 1456 1456 +1 5 4 6 +1 5 4 6 2 02 Addition 37

Solved Step 3: Add the hundreds. Step 4: Add the thousands. Th H T O Th H T O 111 111 1456 1456 +1 5 4 6 +1 5 4 6 002 3002 Th H T O Solve these Th H T O Th H T O 17 5 8 459 2 +5 6 6 2 267 8 +1 4 5 6 +1 3 3 2 38

Concept 5.2: Subtract 3-digit and 4-digit Numbers Think The given grid shows the number of men and women in Farida’s town in the years 2017 and 2018. Years 2017 2018 How can Farida find out how may more Men 2254 2187 men than women lived in her town in the Women 2041 2073 two years. Recall Recall that we can subtract numbers by writing the smaller number below the greater number. A 2-digit number can be subtracted from a larger 2-digit number or a 3-digit number. Similarly, a 3-digit number can be subtracted from a larger 3-digit number. Let us answer these to revise the concept. a) 15 – 0 = _________ b) 37 – 36 = _________ c) 93 – 93 = _________ f) 50 – 45 = _________ d) 18 – 5 = _________ e) 47 – 1 = _________ & Remembering and Understanding We have learnt how to subtract two 3-digit numbers without regrouping. Let us now learn how to subtract them with regrouping. Subtract 3-digit numbers with regrouping When a larger number is to be subtracted from a smaller number, we regroup the next higher place and borrow. And, we always start subtracting from the ones place. Let us understand this with an example. Example 6: Subtract 427 from 586. Solution: To subtract, write the smaller number below the larger number. 48

Step 1: Subtract the ones. But, 6 – 7 is Solved Step 3: Subtract the not possible as 6 < 7. So, regroup the hundreds. digits in the tens place. Step 2: Subtract the tens. 8 tens = 7 tens + 1 tens. Borrow 1 ten to the ones place. Reduce the tens by 1 ten. Now subtract 7 ones from 16 ones. H TO H TO H TO 7 16 7 16 5 5 7 16 –4 8\\ 6\\ 5 \\8 \\6 –4 \\8 \\6 27 –4 2 7 1 9 59 27 59 H TO Solve these H TO H TO 6 23 5 52 4 53 – 3 76 – 2 63 – 2 64 Subtract 4-digit numbers without regrouping Subtracting a 4-digit number from a larger 4-digit number is similar to subtracting a 3-digit number from a larger 3-digit number. The following examples help you understand this better. Example 7: Subtract: 5032 from 7689 Solution: To subtract, write the smaller number below the larger number. Solved Step 1: Subtract the ones. Step 2: Subtract the tens. Th H T O Th H T O 76 8 9 76 8 9 −50 3 2 −50 3 2 7 5 7 Subtraction 49

Step 3: Subtract the hundreds. Solved Step 4: Subtract the thousands. Th H T O Th H T O 7689 7 68 9 −5032 − 5 03 2 2 65 7 657 Solve these Th H T O Th H T O Th H T O 2879 4789 8000 –2137 –2475 –2000 Subtract 4-digit numbers with regrouping In subtraction of 4-digit numbers, we can regroup the digits in thousands, hundreds and tens places. Let us see an example. Example 8: What is the difference 7437 and 4868? Solution: Write the smaller number below the larger number. Steps Solved Solve these Step 1: Subtract the ones. Th H T O Th H T O But, 7 − 8 is not possible as 1654 74 2 17 −1 2 4 6 7 < 8. So, regroup the tens digit, −4 8 3. 3 tens = 2 tens + 1 ten. Borrow 3\\ \\7 1 ten to the ones place. 6 8 9 Step 2: Subtract the tens. But, Th H TO 12 2 − 6 is not possible as 2 < 6. 7 So, regroup the hundreds digit, −4 3 \\2 17 4. 4 hundreds = 3 hundreds + 4\\ 3\\ \\7 1 hundred. Borrow 1 hundred to 868 the tens place. 69 50

Steps Solved Solve these Th H T O Step 3: Subtract the hundreds. Th H T O But, 3 − 8 is not possible. So, 13 12 5674 regroup the thousands digit, −2 3 8 2 7. 7 thousands = 6 thousands + 6 \\3 \\2 17 1 thousand. Borrow 1 thousand \\7 4\\ 3\\ \\7 to the hundreds place. −4 8 6 8 569 Step 4: Subtract the thousands. Th H T O Th H T O 13 12 7468 6 \\3 \\2 17 −4 8 3 7 \\7 4\\ 3\\ \\7 −4 8 6 8 2569 Subtraction 51

Chapter Multiplication 6 Let Us Learn About • using repeated addition to construct multiplication tables. • multiplying 2-digit numbers with and without regrouping. • doubling the numbers mentally. Concept 6.1: Multiply 2-digit Numbers Think Farida bought 2 boxes of toffees to distribute among her classmates on her birthday. Each box has 25 toffees inside it. If there are 54 students in her class, do you think she has enough toffees? Recall In Class 2, we have learnt that multiplication is repeated addition. The symbol ‘×’ indicates multiplication. Multiplication means having a certain number of groups of the same size. 59

Let us recall the multiplication tables of numbers from 1 to 6. 1 2 3 1×1=1 2×1=2 3×1=3 1×2=2 2×2=4 3×2=6 1×3=3 2×3=6 3×3=9 1×4=4 2×4=8 3 × 4 = 12 1×5=5 2 × 5 = 10 3 × 5 = 15 1×6=6 2 × 6 = 12 3 × 6 = 18 1×7=7 2 × 7 = 14 3 × 7 = 21 1×8=8 2 × 8 = 16 3 × 8 = 24 1×9=9 2 × 9 = 18 3 × 9 = 27 1 × 10 = 10 2 × 10 = 20 3 × 10 = 30 4 5 6 4×1=4 5×1=5 6×1=6 4×2=8 5 × 2 = 10 6 × 2 = 12 4 × 3 = 12 5 × 3 = 15 6 × 3 = 18 4 × 4 = 16 5 × 4 = 20 6 × 4 = 24 4 × 5 = 20 5 × 5 = 25 6 × 5 = 30 4 × 6 = 24 5 × 6 = 30 6 × 6 = 36 4 × 7 = 28 5 × 7 = 35 6 × 7 = 42 4 × 8 = 32 5 × 8 = 40 6 × 8 = 48 4 × 9 = 36 5 × 9 = 45 6 × 9 = 54 4 × 10 = 40 5 × 10 = 50 6 × 10 = 60 Let us now construct multiplication tables of 7, 8 and 9. We can then learn to multiply 2-digit numbers. & Remembering and Understanding In multiplication of two numbers: • The number written to the left of the ‘×’ sign is called the multiplicand. • The number written to the right of the ‘×’ sign is called the multiplier. • The number written to the right of the ‘=’ sign is called the product. 60

Multiplication Fact ↓↓ ↓ Multiplicand Multiplier Product Note: (a) R epresenting the multiplicand, multiplier and product using the symbols ‘×’ and ‘=’ is called a multiplication fact. (b) The multiplicand and the multiplier are also called the factors of the product. (c) The product is also called the multiple of both the multiplicand and the multiplier. For example, 2 × 7 = 14 = 7 × 2; 4 × 5 = 20 = 5 × 4 and so on. Order Property: Changing the order in which the numbers are multiplied does not change the product. This is called order property of multiplication. Using multiplication facts and order property, let us now construct the multiplication tables of 7, 8 and 9. 7 8 9 7×1=7 8×1=8 9×1=9 7 × 2 = 14 8 × 2 = 16 9 × 2 = 18 7 × 3 = 21 8 × 3 = 24 9 × 3 = 27 7 × 4 = 28 8 × 4 = 32 9 × 4 = 36 7 × 5 = 35 8 × 5 = 40 9 × 5 = 45 7 × 6 = 42 8 × 6 = 48 9 × 6 = 54 7 × 7 = 49 8 × 7 = 56 9 × 7 = 63 7 × 8 = 56 8 × 8 = 64 9 × 8 = 72 7 × 9 = 63 8 × 9 = 72 9 × 9 = 81 7 × 10 = 70 8 × 10 = 80 9 × 10 = 90 Multiply 2-digit numbers by 1-digit numbers Now, let us learn to multiply a 2-digit number by a 1-digit number. Consider the following example. Multiplication 61

Example 1: Find the product: 23 × 7 Solution: Follow these steps to find the product. Steps Solved Solve these Step 1: Multiply the ones. 3 × 7 = 21 H TO Step 2: Regroup the product. 21 ones = 2 tens and 1 ones 17 Step 3: Write the ones digit of ×9 the product in the product TO and carry over the tens digit 2 H TO to the tens place. 23 15 ×7 ×4 Step 4: Multiply the tens. Step 5: Add the carry over 1 from step 3 to the product. Step 6: Write the sum in the 2 × 7 = 14 tens place. 14 + 2 = 16 H TO 2 23 ×7 161 62

Concept 6.2: Multiply 3-digit Numbers by 1-digit and 2-digit Numbers Think Farida collected some shells and put them into 9 bags. If each bag has 110 shells, how many shells did she collect? Recall We have learnt to multiply a 2-digit number with a 1-digit number. We have also learnt to regroup the ones in multiplication. Multiplication 63

Let us answer these to revise the concept. a) 22 × 2 = _________ d) 33 × 4 = _________ b) 42 × 1 = _________ e) 50 × 2 = _________ c) 11 × 7 = _________ f) 45 × 3 = _________ & Remembering and Understanding We multiply 3-digit numbers just as we multiply 2-digit numbers. Multiply 3-digit numbers by 1-digit numbers without regrouping Let us understand the step-by-step procedure through a few examples. Example 6: Multiply: 401 × 3 Solution: Follow these steps to multiply the given numbers. Step 1: Multiply the ones Solved Step 3: Multiply the hundreds Step 2: Multiply the tens H TO Th H T O 401 H TO 401 401 ×3 ×3 3 ×3 1203 03 H TO Solve these H TO 220 232 HTO ×4 13 0 ×3 ×2 Multiply 3-digit numbers by 1-digit numbers with regrouping We always start multiplying the ones of the multiplicand by the ones of the multiplier. When a 3-digit number is multiplied by a 1-digit number, we may get a 2-digit product in any or all of the places. We regroup these products and carry over the tens digit of the product to the next place. Let us understand this better through an example. 64

Example 7: Multiply: 513 × 5 Solution: Follow these steps to multiply the given numbers. Steps Solved Solve these H TO Step 1: Multiply the ones and write the H TO product under ones. Regroup if the 444 product has two or more digits. 1 3 ×8 5 51 5 × Step 2: Multiply the tens. Add the carry H TO H TO over (if any) to the product. Write the sum under tens. 1 342 ×5 Regroup if the product has two or more 513 digits. ×5 65 Step 3: Multiply the hundreds. Add the Th H T O H TO carry over (if any) to the product and write the sum under hundreds. Regroup if 1 635 the product has two or more digits. ×7 513 ×5 2 565 Multiply 3-digit numbers by 2-digit numbers Multiplication of 3-digit numbers by 2-digit numbers may sometimes involve regrouping too. Let us understand this concept through step-by-step procedure. Consider the following examples. Example 8: Multiply: 243 × 34 Solution: Follow these steps to multiply the given numbers. Multiplication 65

Steps Solved Solve these Step 1: Arrange the numbers in columns, H TO as shown. H TO 141 243 ×22 Step 2: Multiply the ones of the ×34 multiplicand by the ones digit of the H TO multiplier. 3 × 4 = 12 H TO 1 453 Write 2 in the ones place of the product. ×13 Write 1 in the tens place as the carry over. 243 ×34 H TO Step 3: Multiply the tens by the ones digit of the multiplier. 4 × 4 = 16 2 263 ×23 Add the carry over from the previous H TO step. So, 16 + 1 = 17. Write 7 in the tens 11 place of the product and 1 in the 243 hundreds place as the carry over. ×34 Step 4: Multiply the hundreds by the ones digit of the multiplier. 2 × 4 = 8 72 Add the carry over from the previous H TO step. So, 8 + 1 = 9. Write 9 in the hundreds 11 place of the product. 243 ×34 Step 5: Write 0 in the ones place. 972 Multiply the ones of the multiplicand by HTO the tens digit of the multiplier. Write the 11 product under the tens place. 243 ×3 4 3×3=9 972 Step 6: Multiply the tens by the tens digit 90 of the multiplier. H TO 4 × 3 = 12 1 Write 2 in the hundreds place of the 11 product and 1 in hundreds place of the 243 multiplicand as the carry over. ×34 972 290 66

Steps Solved Solve these Step 7: Multiply the hundreds by the tens digit of the multiplier. Th H T O H TO 2×3=6 1 352 ×23 Add the carry over from the previous 11 step. So, 6 + 1 = 7. Write 7 in the thousands 243 place of the multiplicand. ×34 972 7290 Step 8: Add the products and write the Th H T O sum. The sum is the required product. 1 11 243 ×34 972 7290 8262 Multiplication 67

Contents 3Class Part 2 13 19 8 Division 29 8.1 Division as Equal Grouping 8.2 Divide 2-digit and 3-digit Numbers by 1-digit Numbers 9 Fractions 9.1 Fraction as a Part of a Whole

Chapter Division 8 Let Us Learn About • equal grouping and sharing. • repeated subtraction and division facts. • dividing 2-digit number by 1-digit number. • checking the correctness of division. Concept 8.1: Division as Equal Grouping Think Farida and Piyush got a chocolate bar with 14 pieces for Christmas. Piyush divided it and gave Farida 6 pieces. Do you think Farida got an equal share? How can we find out? Recall In the previous chapter, we have learnt multiplication. Multiplication is finding the total number of objects that have been grouped equally. Let us use this to distribute objects equally in groups. Consider 12 bars of chocolate. The different ways in which they can be distributed are as follows. 13

Distributing in 1 group: 1 × 12 = 12 Distributing in 2 groups: 2 × 6 = 12 Distributing in 3 groups: 3 × 4 = 12 Distributing in 4 groups: 4 × 3 = 12 Distributing in 6 groups: 6 × 2 = 12 Distributing in 12 groups: 12 × 1 = 12 Distributing a given number of objects into equal groups is called division. We can understand division better by using equal sharing and equal grouping. & Remembering and Understanding Equal sharing means having equal number of objects or things in a group. We use division to find the number of things in a group and the number of groups. 14

Suppose 9 balloons are to be shared 1st round: 1 balloon is taken by each equally among 3 friends. Let us use friend. repeated subtraction to distribute the balloons. 9 – 3 = 6. So, 6 balloons remain. 2nd round: From the remaining 6 balloons, 3rd round: From the remaining 3 balloons, 1 more balloon is taken by each friend. 1 more balloon is taken by each friend. Now, each friend has 2 balloons. Now, each of them has 3 balloons. 6 – 3 = 3. So, 3 balloons remain. 3 – 3 = 0. So, 0 balloons remain. Each friend gets 3 balloons. We can write it as 9 divided by 3 equals 3. 9 divided by 3 equals 3 is written as ↓ ↓ ↓ Total Number of Number of number of objects in each groups objects group Quotient Dividend Divisor Division 15

In a division, the number that is divided is called the dividend. The number that divides is called the divisor. The answer in division is called the quotient. The number (part of the dividend) that remains is called the remainder. The symbol for ‘is divided by’ is ÷. 9 ÷ 3 = 3 is called a division fact. In this, 9 is the dividend, 3 is the divisor and 3 is the quotient. Note: Representing the dividend, divisor and quotient using the symbols ÷ and = is called a division fact. We use multiplication tables to find the quotient in a division. We find the factor which when multiplied by the divisor gives the dividend. Let us understand this through a few examples. Example 1: 18 pens are to be shared equally by 3 children. How many pens does each of them get? Solution: Total number of pens = 18 Number of children = 3 Number of pens each child gets = 18 ÷ 3 = 6 (since 6 × 3 = 18) Therefore, each child gets 6 pens. Example 2: 10 flowers are put in some vases. If each vase has 2 flowers, how many vases are used? Solution: Number of flowers = 10 Number of flowers in each vase = 2 Number of vases used = 10 ÷ 2 = 5 (since 5 × 2 = 10) Therefore, 5 vases are used to put 10 flowers. We get two division facts from a multiplication fact. The divisor and the quotient are the factors of the dividend. 16

Observe the following table: Dividend ÷ Divisor = Quotient Multiplicand × Multiplier = Product 6 × 3 = 18 18 ÷ 6 = 3 ↓↓ ↓ ↓↓ ↓ Divisor Quotient Dividend Product Factor Factor (Multiplicand) (Multiplier) From the multiplication fact 6 × 3 = 18, we can write two division facts: a) 18 ÷ 3 = 6 and b) 18 ÷ 6 = 3 Multiplication and division are reverse operations. Let us now understand this through an activity. We can show a multiplication fact on the number line. For example, 5 × 3 = 15 means 5 times 3 is 15. To show 5 times 3 on the number line, we take steps of 3 for 5 times. We go forward from 0 to 15. Similarly, we can show the division fact 15 ÷ 3 = 5 on the number line. To show 15 divided by 3 on the number line, we take steps of 3 for 5 times. We go backward from 15 to 0 as shown. Division 17

Concept 8.2: Divide 2-digit and 3-digit Numbers by 1-digit Numbers Think Farida has 732 stickers. She wants to distribute them equally among her three friends. How will she distribute? Recall In the previous section, we have learnt that division is related to multiplication. For every division fact, we can write two multiplication facts. For example, the two multiplication facts of 35 ÷ 7 = 5 are: a) 7 × 5 = 35 and b) 5 × 7 = 35. Let us answer these to recall the concept of division. a) The number which divides a given number is called _________________. b) T he answer we get when we divide a number by another is called ______________________. c) T he division facts for the multiplication fact 2 × 4 = 8 are ________________ and __________________. Division 19

& Remembering and Understanding We can make equal shares or groups and divide with the help of vertical arrangement. A number divided by the same number is always 1. Let us see some examples. Dividing a 2-digit number by a 1-digit number (1-digit quotient) Example 7: Solve: 45 ÷ 5 Solution: Follow these steps to divide a 2-digit number by a 1-digit number. Steps Solved Solve these Step 1: Write the dividend and 5)45 Dividend = _____ Divisor = ______ )divisor as shown: Divisor Dividend Quotient = ____ Remainder = _____ Step 2: Find the multiplication fact 45 = 5 × 9 8) 56 which has the dividend and divisor. - Step 3: Write the other factor as the 9 quotient. Write the product of the factors below the dividend. 5)45 − 45 Step 4: Subtract the product 9 4) 36 Dividend = _____ from the dividend and write the Divisor = ______ difference below the product. 5)45 - Quotient = ____ This difference is called the Remainder = _____ remainder. − 45 00 45 = Dividend 5 = Divisor 9 = Quotient 0 = Remainder Note: If the remainder is zero, the divisor is said to divide the dividend exactly. Checking for correctness of division: The multiplication fact of the division is used to check its correctness. Step 1: Compare the remainder and divisor. The remainder must always be less than the divisor. 20

Step 2: Check if (Quotient × Divisor) + Remainder = Dividend Let us now check if our division in example 7 is correct or not. Step 1: Remainder < Divisor 0 < 5 (True) Step 2: Quotient × Divisor 9×5 Step 3: (Quotient × Divisor) + Remainder = Dividend 45 + 0 = 45 = Dividend Note: The division is incorrect if: a) Remainder > or = divisor b) (Quotient × Divisor) + Remainder ≠ Dividend 2-digit quotient In the examples we have seen so far, the quotients are 1-digit numbers. In some divisions, the quotients may be 2-digit numbers. Let us see some examples. Example 8: Solve: 57 ÷ 3 Solution: Follow these steps to divide a 2-digit number by a 1-digit number. Steps Solved Solve these Step 1: Check if the tens digit of the dividend is greater than the divisor. 5>3 5) 60 Step 2: Divide the tens and write the quotient. 1 − Write the product of quotient and divisor, below the tens digit of the dividend. 3)57 − Step 3: Subtract and write the difference −3 Step 4: Check if difference < divisor is true. 1 Dividend = _____ Divisor = ______ 3)57 Quotient = ____ Remainder = ___ −3 2 2 < 3 (True) Division 21

Steps Solved Solve these Step 5: Bring down the ones digit of the 1 3) 42 dividend and write it beside the remainder. 3)57 − − − 3↓ 27 Step 6: Find the largest number in the 3 × 8 = 24 1 multiplication table of the divisor that can be subtracted from the 2-digit number in )3 × 9 = 27 3 57 the previous step. 3 × 10 = 30 24 < 27 < 30. − 3↓ So, 27 is the 27 required number. Step 7: Write the factor of required number, 19 Dividend = _____ other than the divisor, as the quotient. Write Divisor = ______ the product of the divisor and the quotient 3)57 Quotient = ____ below the 2-digit number. Subtract and Remainder = ___ write the difference. − 3↓ 27 Step 8: Check if remainder < divisor is true. Stop the division. − 27 00 0 < 3 (True) (If this is false, the division is incorrect.) RQeumotaieinndte=Trr1=a90in My Brain Step 9: Write the quotient and the remainder. Step 10: Check if (Divisor × Quotient) + 3 × 19 + 0 = 57 Remainder = Dividend is true. 57 + 0 = 57 57 = 57 (True) (If this is false, the division is incorrect.) Divide 3-digit numbers by 1-digit numbers (2-digit quotient) Dividing a 3-digit number by a 1-digit number is similar to dividing a 2-digit number by a 1-digit number. Let us understand this through a few examples. Example 9: Solve: a) 265 ÷ 5 Solution: Follow these steps to divide a 3-digit number by a 1-digit number. 22

Steps Solved Solve these Step 1: Check if the hundreds digit of 4) 244 the dividend is greater than the divisor. 5)265 − If it is not, consider the tens digit too. 2 is not greater than 5. So, consider 26. Step 2: Find the largest number that 5 − can be subtracted from the 2-digit number of the dividend. Write the 5)265 Dividend = _____ quotient. Divisor = ______ − 25 Quotient = ____ Remainder = ___ Write the product of the quotient and 5 × 4 = 20 the divisor below the dividend. 5 × 5 = 25 9) 378 5 × 6 = 30 Step 3: Subtract and write the − difference. 25 < 26 − 5 5)265 − 25 1 Step 4: Check if difference < divisor 1 < 5 (True) is true. (If it is false, the division is incorrect.) Step 5: Bring down the ones digit 5 of the dividend. Write it beside the remainder. 5)265 − 25↓ 15 Step 6: Find the largest number in the 5 multiplication table of the divisor that can be subtracted from the 2-digit 5)265 number in the previous step. − 25↓ 15 5 × 2 = 10 5 × 3 = 15 5 × 4 = 20 15 is the required number. Division 23

Steps Solved Solve these Step 7: Write the factor of required 53 number, other than the divisor, as Dividend = _____ quotient. Write the product of divisor 5)265 Divisor = ______ and quotient below the 2-digit Quotient = ____ number. Then, subtract them. − 25↓ Remainder = ___ 15 Step 8: Check if remainder < divisor is true. Stop the division. (If this is false, − 15 the division is incorrect.) 00 0 < 5 (True) Step 9: Write the quotient and Quotient = 53 remainder. Remainder = 0 Step 10: Check if (Divisor × Quotient) + 5 × 53 + 0 = 265 Remainder = Dividend is true. (If this is 265 + 0 = 265 false, the division is incorrect.) 265 = 265 (True) 3-digit quotient Example 10: Solve: 784 by 7 Solution: Follow these steps to divide a 3-digit number by a 1-digit number. Steps Solved Solve these Step 1: Check if the hundreds digit of the dividend is greater than or equal to the 7)784 8) 984 divisor. Step 2: Divide the hundreds and write the 7=7 − quotient in the hundreds place. 1 − Write the product of the quotient and the divisor under the hundreds place of the 7)784 − dividend. −7 24

Steps Solved Solve these Step 3: Subtract and write the difference. Dividend = _____ 1 Divisor = ______ Quotient = ____ 7)784 Remainder = ___ −7 5) 965 0 − Step 4: Check if difference < divisor is true. 0 < 7 (True) Step 5: Bring down the next digit of the 1 − dividend. Check if it is greater than or − equal to the divisor. 7)7 84 Dividend = _____ − 7↓ Divisor = ______ 08 Quotient = ____ 8>7 Remainder = ___ Step 6: Find the largest number in the 11 multiplication table of the divisor that can be subtracted from the 2-digit number in 7)784 the previous step. − 7↓ Write the factor other than the divisor as 08 quotient. −7 Write the product of the quotient and the divisor below it. 7×1=7<8 The required number is 7. Step 7: Subtract and write the difference. 11 Bring down the next digit (ones digit) of the dividend. 7)784 Check if the dividend is greater than or − 7↓ equal to the divisor. 08 −7 14 14 > 7 Division 25

Steps Solved Solve these Step 8: Find the largest number in the multiplication table of the divisor that can 112 2) 246 be subtracted from the 2-digit number in the previous step. 7)784 − Write the factor other than the divisor as − the quotient. − 7↓ − Write the product of the quotient and the 08 divisor below it. Dividend = _____ −7 Divisor = ______ Step 9: Subtract and write the difference. 14 Quotient = ____ Check if it is less than the divisor. Stop the Remainder = ___ division. − 14 Step 10: Write the quotient and the 7 × 2 = 14 remainder. The required Step 11: Check if (Divisor × Quotient) + number is 14. Remainder = Dividend is true. (If it is false, the division is incorrect.) 112 7)784 − 7↓ 08 −7 14 − 14 00 Quotient = 112 Remainder = 0 7 × 112 + 0 = 784 784 + 0 = 784 784 = 784 (True) 26

Chapter Fractions 9 Let Us Learn About • fractions as a part of a whole and their representation. • identify parts of fractions. • fractions of a collection. • applying the knowledge of fractions in real life. Concept 9.1: Fraction as a Part of a Whole Think Farida and her three friends, Joseph, Salma and Rehan, went on a picnic. Farida had only one apple with him. He wanted to share it equally with everyone. What part of the apple does each of them get? Recall Look at the rectangle shown below. We can divide the whole rectangle into many equal parts. Consider the following: 29

1 part: 2 equal parts: 3 equal parts: 4 equal parts: 5 equal parts: and so on. Let us understand the concept of parts of a whole through an activity. & Remembering and Understanding Suppose we want to share an apple with our friends. First, we count our friends with whom we want to share the apple. Then, we cut it into as many equal pieces as the number of persons. Thus, each person gets an equal part of the apple after division. Parts of a whole A complete or full object is called a whole. Observe the following parts of a chocolate bar: whole 2 equal parts 3 equal parts 4 equal parts We can divide a whole into equal parts as shown above. Each such division has a different name. To understand this better, let us do an activity. 30

Activity: Halves Take a square piece of paper. Fold it into two equal parts as shown. Each of the equal parts is called a ‘half’. ‘Half’ means 1 out of 2 equal parts. Putting these 2 equal parts together makes the complete piece of paper. We write ‘1 out of 2 equal parts’ as 1 . 2 In 1 , 1 is the number of parts taken and 2 is the total number of equal parts the whole 2 is divided into. Note: 1 and 1 make a whole. 2 2 Thirds In figure (a), observe that the three parts are not equal. We can also divide a whole into three equal parts. Fold a rectangular piece of paper as shown in figures (b) and (c). 11 1 33 3 three parts three equal parts Fig. (c) Fig. (a) Fig. (b) Each equal part is called a third or one-third. The shaded part in figure (c) is one out of three equal parts. So, we call it a one-third. Two out of three equal parts of figure (c) are not shaded. We call it two-thirds (short form of 2 one-thirds). We write one-third as 1 and two-thirds as 2 . 3 3 Note: 1 , 1 and 1 or 1 and 2 makes a whole. 3 3 3 3 3 Fractions 31

Fourths Similarly, fold a square piece of paper into four equal parts. Each of them is called a fourth or a quarter. In figure (d), the four parts are not equal. In figure (e), each equal part is called a fourth or a quarter and is written as 1 . 4 1 Four parts 4 Fig. (d) 1 4 1 4 1 4 Four equal parts Fig. (e) Two out of four equal parts are called two-fourths and three out of four equal parts are called three-fourths, written as 2 and 3 respectively. 44 Note: Each of 1 and 3 ; 1 , 1 , 1 and 1 and 1 , 1 and 2 make a whole. 4 4 4 4 4 4 4 4 4 The total number of equal parts a whole is divided into is called the denominator. The number of such equal parts taken is called the numerator. Representing the parts of a whole as Numerator is called a fraction. Numbers of Denominator the form Numerator are called fractions. Thus, a fraction is a part of a whole. Denominator 32

For example, 1 , 1, 1, 2 and so on are fractions. 2 3 4 3 Let us now see a few examples. Example 1: Identify the fraction for the shaded parts in the figures below. a) b) Solution: Steps Solved Solve this a) b) Step 1: Count the number of equal parts, the figure is divided into Total number of Total number of equal (Denominator). parts = _______ equal parts = 8 Number of parts shaded Step 2: Count the number of Number of parts = ______ shaded parts (Numerator). shaded = 5 Step 3: Write the fraction Fraction = 5 Numerator . 8 Denominator Fraction = Example 2: The circular disc shown in the figure is divided into equal parts. What fraction of the disc is painted yellow? Write the fraction of the disc that is painted white. Solution: Total number of equal parts of the disc is 16. The fraction of the disc that is painted yellow = Number of parts painted yellow = 3 Total number of equal parts 16 The fraction of the disc that is painted white = Number of parts painted white = 7 Total number of equal parts 16 Fractions 33

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