Higher Order Thinking Skills (H.O.T.S.) We can check the correctness of a subtraction problem using addition. Consider an example. b) 145 from 364. Example 8: Subtract: a) 27 from 36 Solution: a) 36 – 27 b) 364 – 145 TO HT O 2 16 5 14 \\3 \\6 3 \\6 4\\ −2 7 −1 4 5 9 21 9 36 – 27 = 9 364 – 145 = 219 We can write 36 = 27 + 9 We can write 364 = 145 + 219 We can conclude that to check if the subtraction is correct, we add the subtrahend (the number being subtracted) and the difference. If this sum is the same as the minuend (the number from which a number is subtracted), the subtraction is correct. Framing word problems Let us consider these subtraction facts. a) 37 – 14 = 23 b) 37 – 23 = 14 We can try to frame some interesting situations and problems using these subtraction facts. a) Of the 37 students in class, 14 are in the green house. How many students are in the red house? b) 37 children are playing on the ground. 23 of them are playing football. How many are playing basketball? Similarly, we can frame some interesting problems using subtraction facts of 3-digit numbers. Let us see an example. Example 9: Frame a word problem using: a) 706 – 234 = 472 b) 461 − 110 = 351 Solution: A few possible word problems are: a) In a school, there are 706 students. 234 students were absent on Monday. How many students were present? b) 4 61 people booked a train for a trip to Goa. 110 people cancelled the trip. How many people went on the trip? Subtraction 47 Hazelwood_Maths_G03_TB_Part 1.indb 47 05-08-2023 21:36:28
Drill Time Concept 5.2: Subtract 3-digit and 4-digit Numbers 1) Subtract 3-digit numbers with regrouping. a) 254 – 173 b) 678 – 619 c) 147 – 129 d) 781 – 682 e) 356 – 177 f) 543 -267 2) Subtract 4-digit numbers without regrouping. a) 2341 – 1230 b) 7632 – 5120 c) 9763 – 2311 d) 7629 – 1318 e) 7589 – 1268 f) 8657 - 4320 3) Subtract 4-digit numbers with regrouping. a) 7632 – 1843 b) 4391 – 2482 c) 9843 – 7943 d) 8325 – 5436 e) 6893 – 3940 f) 9746 - 5859 4) Word problems a) A stick is 8745 cm long. A 4313 cm long piece is cut from it. What length of the stick is remaining? b) R aj stays 5786 m away from Matin’s house. Raj travels 3825 m of the distance. What is the distance yet to be covered by Raj to reach Matin’s house? c) A shopkeeper has 481 juice bottles. If he sells 177 bottles in a week, find the number of remaining bottles. d) 3 72 students participated in the annual day celebration of a school. If 199 of them were boys, find the number of girl participants. e) A van transported 6454 milk packets on a particular day. 3224 milk packets were sent to shop A and the remaining to shop B. How many packets were sent to shop B? 48 05-08-2023 21:36:28 Hazelwood_Maths_G03_TB_Part 1.indb 48
5.3: Subtract 2-digit Numbers Mentally Do you recall how to subtract two 1-digit numbers mentally? If yes, first try to describe the steps in your own words. When we subtract two 1-digit numbers, we keep the larger of the two numbers in mind and then subtract the smaller one from it. We can apply the same principle when mentally subtracting 2-digit numbers as well. How do we subtract 2-digit numbers mentally? Let’s learn this concept with the help of an example. Suppose you have to subtract 35 from 69, here are the steps: 1) Mentally subtract the digits in the ones place of the two numbers: 9 – 5 = 4. 2) Then, mentally subtract the digits in the tens place of the two numbers: 6 – 3 = 3. 3) Now write the differences obtained in steps 1) and 2) as the difference between the two numbers; so, 69 – 35 = 34. Do you know how to mentally subtract numbers when the digit in the ones place of the smaller number is more than the digit in the ones place of the larger number like in the case of 83 – 46? Try and remember the concept of regrouping and discuss with your partner hnow you would use it to mentally subtract 46 from 83? How do we subtract 2-digit numbers mentally by regrouping? Suppose you want to mentally subtract 29 from 56. Follow the steps below. 1) Regroup the numbers as tens and ones. So 29 = 20 + 9 and 56 = 50 + 6. 2) Notice that we cannot subtract the digits in the ones place; 6 – 9 is not possible because 6 < 9. So, regroup the digit in the tens place: 5 tens = 4 tens + 1 ten. 3) Borrow the 1 ten to the 6 in the ones place: 10 + 6 = 16. 4) Now, subtract the digits in the ones place of the two numbers mentally. Use regrouping again if needed: 16 – 9 = 7. Subtraction 49 Hazelwood_Maths_G03_TB_Part 1.indb 49 05-08-2023 21:36:29
5) Then, subtract the digits in the tens place of the two numbers mentally: 4 tens – 2 tens = 2 tens. 6) Write down the answers from the steps 4) & 5) together as the final difference: 2 tens + 7 ones = 20 + 7 = 27. 7) So, 56 – 29 = 27 Alisha went to school for 49 days in Term I and 65 days in Term II. For how many more days did Alisha go to school in Term II than in Term I? Solve mentally. C an you think of a few similar problems using 3-digit or 4-digit numbers and subtract them mentally? Work on it with your partner and share it with the class. Reflection Time! 1) Imagine that your father gave you `1000 to buy your favourite comic books. When you are visiting the comic exhibition, subtract mentally the cost of each comic book that you would like to buy from the `1000 your father gave you. How many comic books would you be able to buy in all? 2) If you can’t visit a comic exhibition, find out the approximate cost of comic books from your parents and do the calculation mentally. 3) Why is regrouping important ? Draw the place value chart up to 4 places and explain regrouping to your friends. Drill Time Concept 5.3: Subtract 2-digit Numbers Mentally 1) Subtract 2-digit numbers mentally without regrouping. a) 43 from 84 b) 24 from 76 c) 52 from 87 d) 34 from 75 e) 14 from 38 f) 46 from 98 50 05-08-2023 21:36:30 Hazelwood_Maths_G03_TB_Part 1.indb 50
Drill Time 2) Subtract 2-digit numbers mentally with regrouping. a) 42 from 81 b) 28 from 84 c) 11 from 20 d) 23 from 51 e) 76 from 81 f) 35 from 73 3) Word problems a) Rehmat has 48 pencils. He has used 29 pencils. How many pencils are left with him? b) S am travelled for 23 km on Day 1 and 76 km on Day 2. How much more distance (in km) did Sam travel on Day 2 than on Day 1? c) There are 36 monkeys in a forest, of which 18 are male. How many female monkeys are there in a forest? d) J ay has 69 oranges. He gives 42 oranges to Raj. How many oranges are left with Jay? Connect the Dots English Fun Let us read a poem to learn subtraction. More on top? No need to stop! More on the floor? Go next door and get 10 more! Number the same? Zero’s the game! Subtraction 51 Hazelwood_Maths_G03_TB_Part 1.indb 51 05-08-2023 21:36:31
Chapter Multiplication 6 Let Us Learn About • using multiplication facts and order property 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. 52 05-08-2023 21:36:33 Hazelwood_Maths_G03_TB_Part 1.indb 52
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. Multiplication 53 Hazelwood_Maths_G03_TB_Part 1.indb 53 05-08-2023 21:36:33
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. 54 05-08-2023 21:36:36 Hazelwood_Maths_G03_TB_Part 1.indb 54
Example 1: Find the product: 23 × 7 Solution: Follow these steps to find the product. Steps Solved Solve these 3 × 7 = 21 H TO Step 1: Multiply the ones. 21 ones = 2 tens and 1 ones Step 2: Regroup the product. 17 Step 3: Write the ones digit of TO ×9 the product in the product and carry over the tens digit 2 H TO to the tens place. 23 15 ×7 ×4 1 Step 4: Multiply the tens. 2 × 7 = 14 Step 5: Add the carry over 14 + 2 = 16 from step 3 to the product. H TO Step 6: Write the sum in the 2 23 tens place. ×7 161 Application Let us now see some real-life situations where we use multiplication of 2-digit numbers. Example 2: There were 54 students in a class of a school. The school had 8 such classes. How many students were there in the entire school? H T O Solution: Number of students in one class = 54 students 3 Number of classes in the school = 8 54 Number of students in the school = 54 × 8 ×8 Therefore, the total number of students in the school = 432 4 3 2 Example 3: Manoj travelled 7 km in a day. If he travels the same distance every day, how much distance does he travel in 25 days? H TO Solution: The distance that Manoj travelled in a day = 7 km 3 He travels the same distance every day. The distance he 25 travels in 25 days = 25 × 7. ×7 Therefore, Manoj travels 175 km in 25 days. 175 Multiplication 55 Hazelwood_Maths_G03_TB_Part 1.indb 55 05-08-2023 21:36:36
Higher Order Thinking Skills (H.O.T.S.) Let us now try to frame a few multiplication word problems using multiplication. Example 4: Number of chocolates in a box = 9 Number of such boxes = 5 Total chocolates = 45 Solution: Word problem: A box contains 9 chocolates. There are 5 such boxes. Find the total number of chocolates. Example 5: Frame a word problem with the given fact. 8 × 2 = 16 Solution: Word problem: There are 2 rows with 8 students in each row. What is the total number of students? Drill Time Concept 6.1: Multiply 2-digit Numbers 1) Multiply 2-digit numbers by 2, 3, 4, 5 and 6. a) 56 × 3 b) 23 × 2 c) 77 × 6 d) 50 × 5 e) 62 × 4 2) Multiply 2-digit numbers by 7, 8 and 9. a) 23 × 9 b) 12 × 7 c) 76 × 8 d) 84 × 8 e) 83 × 9 3) Word problems a) T here were 23 students in one group. The school had 4 such groups. How many students were there in all the groups? b) V iraj travelled for 30 km in one day. He travelled the same distance everyday for 7 days. How many kilometres did he travel in 7 days? 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? 56 05-08-2023 21:36:37 Hazelwood_Maths_G03_TB_Part 1.indb 56
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. 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 Multiplication 57 Hazelwood_Maths_G03_TB_Part 1.indb 57 05-08-2023 21:36:37
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. 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 1 444 product has two or more digits. ×8 513 ×5 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. 58 05-08-2023 21:36:37 Hazelwood_Maths_G03_TB_Part 1.indb 58
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 Multiplication 59 Hazelwood_Maths_G03_TB_Part 1.indb 59 05-08-2023 21:36:38
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 11 972 7290 8262 Application Let us now solve some word problems that have real-life applications. Example 9: Payal saves `175 per month for a year. How much money will she have Solution: at the end of the year? Th H T O Amount saved by Payal per month = `175 11 Number of months in a year = 12 175 Total money saved in a year = `175 × 12 × 12 Therefore, Payal has `2100 at the end of the year. 11 350 1750 2100 60 Hazelwood_Maths_G03_TB_Part 1.indb 60 05-08-2023 21:36:38
Example 10: Rohan ran 315 m every day for a week. How many metres did he run in that week? Solution: 1 week = 7 days Th H T O Distance run by Rohan in a day = 315 m 13 Distance he ran in a week = 315 m × 7 = 2205 m 315 So, Rohan covered a total distance of 2205 m in one × 7 week. 2205 Higher Order Thinking Skills (H.O.T.S.) Sometimes, we can find numbers that satisfy two or more conditions. Let us now see a few examples. Example 11: Find two numbers whose sum is 13 and product is 6 more than 30. Solution: The two conditions in this problem are: a) The sum of the numbers is 13 b) The product of the numbers is 6 more than 30 From condition b), 6 more than 30 = 30 + 6 = 36. So, the product of the numbers is 36. Now, let us find the two numbers whose product is 36 and sum is 13. 36 = 1 × 36; 36 = 2 × 18; 36 = 3 × 12; 36 = 4 × T9raanidn36M=y6 ×B6r.aOinf these, the numbers whose sum is 13 are 9 and 4 (since 9 + 4 = 13). Therefore, the required numbers are 9 and 4. Example 12: Find two numbers whose difference is 1 and product is 2 more than 40. Solution: The two conditions in this problem are: a) The difference of the numbers is 1. b) The product of the numbers is 2 more than 40 which is 42. Now, let us find two numbers whose product is 42 and difference is 1. 42 × 1 = 42; 21 × 2 = 42; 14 × 3 = 42; 7 × 6 = 42. Of these, the numbers whose difference is 1 are 7 and 6. Therefore, the required numbers are 7 and 6. Multiplication 61 Hazelwood_Maths_G03_TB_Part 1.indb 61 05-08-2023 21:36:38
Drill Time Concept 6.2: Multiply 3-digit Numbers by 1-digit and 2-digit Numbers 1) Multiply 3-digit numbers by 1-digit number without regrouping. a) 101 × 8 b) 212 × 4 c) 414 × 2 d) 111 × 5 e) 323 × 3 2) Multiply 3-digit numbers by 1-digit numbers (with regrouping). a) 225 × 7 b) 762 × 4 c) 868 × 8 d) 723 × 5 e) 429 × 2 3) Multiply 3-digit numbers by 2-digit numbers. a) 769 × 21 b) 759 × 10 c) 578 × 42 d) 619 × 66 e) 290 × 30 4) Word problems a) Susan drove 462 km every day for a week. What distance did she drive in that week? b) Sohail spends `616 for a set of books. How much will he spend on 24 such sets? 6.3: Double 2-digit and 3-digit Numbers Mentally We have learnt mental addition and subtraction in the previous lessons. Now let us look at how to double a given number mentally. How do we double a 2-digit number? Doubling a number means multiplying the number by 2. To double a number, we must be thorough with the multiplication table of 2 first. For example, 3 × 2 = 6, 5 × 2 = 10, 10 × 2 = 20, and so on. Let us now understand how to double a 2-digit number mentally using an example. Suppose we want to double the number 53. To do so, you can use the following steps. Steps Doubling 53 Double 41 Step 1: Multiply the tens digit by 2. The tens digit is 5. The tens digit is ____. Step 2: If the ones digit is less than or So, 5 × 2 = 10. So, ___ × 2 = ___. equal to 4, write the product in step 1 as it is. If not, add 1 to it and write. The ones digit is 3. The ones digit is ___ 3 < 4 (True) ___ < ___ (True/ False) 62 05-08-2023 21:36:39 Hazelwood_Maths_G03_TB_Part 1.indb 62
Step 3: Multiply the ones digit by 2. 3×2=6 ___ × 2 = ___ 53 × 2 = 106 ___ × 2 = ___ Step 4: Write the products in steps 1 and 3 together. This gives us the double of the given number. 1) Double the given numbers mentally. a) 22 b) 43 c) 51 2) Sony is 36 years old. Her aunt is double her age. How old is Sony’s aunt? Calculate mentally. 3) Rohan bought 42 books in Year 1 and double the number in Year 2. How many books did he buy in Year 2? Calculate mentally. Now, how do we double a 3-digit number? Think of a 3-digit number. Can the same method shown earlier be used? Work on it with your partner and share it with the class. Double the given numbers mentally. 1) 172 2) 530 Reflection Time! 1) Help your friend calculate the total number of chocolates he has to bring to school on his birthday, if he wants to give two chocolates each to all the teachers in the school and all his classmates. 2) Your mother wants to buy exactly the same number of toys, books, clothes and sweets for both you and your sibling. How can you help her get to that number quickly for each of them? Multiplication 63 Hazelwood_Maths_G03_TB_Part 1.indb 63 05-08-2023 21:36:39
Drill Time 6.3: Double 2-digit and 3-digit Numbers Mentally 1) Double the given numbers mentally. a) 23 b) 52 c) 61 d) 120 e) 131 2) Solve word problems mentally. a) S onal earned `28 on Monday. She earned double the amount on Tuesday. How much did she earn on Tuesday? b) A fruitseller sells 38 apples on day 1 and double the apples on day 2. How many apples did he sell on day 2? Connect the Dots Social Studies Fun All the arrangements of Charbagh Garden of Taj Mahal are based on four or its multiples. The entire garden is divided into four parts. There are 16 flowerbeds. It is said that each of the flowerbeds is planted with 400 plants. 64 05-08-2023 21:36:40 Hazelwood_Maths_G03_TB_Part 1.indb 64
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