So, the estimated number of pencils left with Parul = 80 − 30 = 50 Therefore, Parul has about 50 pencils. Example 4: Ram has 94 sweets. He distributes 46 sweets among his friends. About how many sweets remain with Ram? Solution: Number of sweets Ram has = 94 94 rounded off to the nearest tens is 90 (since 4 < 5). Number of sweets distributed among Ram's friends = 46 46 rounded off to the nearest tens is 50 (since 6 > 5). So, the estimated number of sweets left with Ram = 90 − 50 = 40 Therefore, Ram has about 40 sweets. Higher Order Thinking Skills (H.O.T.S.) In some situations, we may need to carry out both addition and subtraction. In such cases, we need to identify which operation is to be carried out first. Example 5: In a school, there are 976 students. Of them, 325 are from the pre-primary section, 416 are from the primary section, and the rest are from high school. How many high school students are there in the school? Solution: Total number of students = 976 HTO Number of students from the pre-primary section = 325 1 Number of students from the primary section = 416 Total number of students in pre-primary and primary 325 school sections = 325 + 416 = 741 +4 1 6 741 Number of students in high school = Total number of HTO students – Number of students in pre-primary and 9 7 6 primary school sections = 976 – 741 = 235 −7 4 1 Therefore, there are 235 high school students. 235 Subtraction 47
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 Application Subtraction of 3-digit numbers is very often used in real life. Here are a few examples. Example 9: Sonu bought 375 marbles. He gave 135 marbles to his brother. Solution: How many marbles are left with him? H TO Total number of marbles Sonu bought = 375 375 −1 3 5 Number of marbles given to Sonu’s brother = 135 Number of marbles left with him = 375 – 135 = 240 2 4 0 Therefore, 240 marbles are left with Sonu. Example 10: Vinod had 536 stamps. He gave some stamps to his brother and then Vinod was left with 278 stamps. How many stamps did Vinod give his Solution: brother? H TO Total number of stamps Vinod had = 536 12 Number of stamps Vinod had after giving some to his brother = 278 4 2\\ 16 \\5 \\3 \\6 Number of stamps he gave his brother = −278 536 – 278 = 258 258 Therefore, Vinod gave 258 stamps to his brother. Subtraction 51
We can use subtraction of 4-digit numbers in real-life situations. Let us see some examples. Example 11: Mohan’s uncle stays 8630 m away from Mohan’s house. Mohan travelled Solution: 6212 m of the distance. What is the distance yet to Th H T O be covered by Mohan to reach his uncle’s house? 2⁄ 1⁄0 Distance between Mohan’s house and his uncle’s 8 630 house = 8630 m − 6 212 Distance travelled by Mohan = 6212 m 2 418 Remaining distance Mohan has to travel = 8630 m – 6212 m = 2418 m Therefore, Mohan has to travel 2418 m more to reach his uncle’s house. Example 12: A rope is 6436 cm long. A 3235 cm long piece is cut from it. How much of the rope is left? Solution: Length of the rope = 6436 cm Th H T O 6436 Length of the piece cut = 3235 cm −3 2 3 5 The length of the remaining piece of rope 3201 = 6436 cm – 3235 cm = 3201 cm Therefore, 3201 cm of the rope is left. 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 13: 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 We can write 36 = 27 + 9 364 – 145 = 219 We can write 364 = 145 + 219 52
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) O f the 37 students in class, 14 are in the green house. How many students are in the red house? b) 3 7 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 14: 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) 461 people booked the train for a trip to Goa. 110 people cancelled the trip. How many people went on the trip? Concept 5.3: Subtract 2-digit Numbers Mentally Think Farida had 19 pens. She gave 12 pens to her sister. Can you find the number of pens remaining with Farida without using a paper and a pencil? Recall Recall that to subtract two 1-digit numbers mentally, we keep the larger number in mind and subtract the smaller one from it. Subtraction 53
Let us answer these to revise the concept. a) 5 – 4 = ________ [] [] (A) 5 (B) 4 (C) 1 (D) 9 [] [] b) 3 – 3 = ________ (C) 0 (D) 5 [] (A) 3 (B) 6 c) 4 – 1 = ________ (A) 3 (B) 4 (C) 6 (D) 8 d) 5 – 0 = ________ (A) 4 (B) 5 (C) 0 (D) 6 e) 6 – 3 = ________ (A) 4 (B) 6 (C) 3 (D) 9 & Remembering and Understanding We have learnt to subtract 1-digit numbers mentally. Let us understand subtraction of 2-digit numbers mentally through an example. Subtract 2-digit numbers mentally without regrouping Example 15: Subtract mentally: 52 from 76 Solution: Follow these steps to subtract mentally. Steps Solved Solve this n 52 from 76 35 from 69 Step 1: Subtract mentally 6 – 2 = 4 ______ – ______ = the digits in the ones place of the two numbers. Step 2: Subtract mentally The digits in the tens place The digits in the tens place of the digits in the tens place of the two numbers are 7 the two numbers are _______ of the two numbers. and 5. and _______. So, imagine that 7 fingers So, imagine that _____ fingers are open. Then imagine are open. closing 5 of them. Then imagine closing ___ of 7–5=2 them. ____– ____ = ___ 54
Steps Solved Solve this 52 from 76 35 from 69 Step 3: Write the difference obtained in So, 76 – 52 = 24. So, 69 – 35 = ____. steps 1 and 2 together as the difference of the given numbers. Sometimes regrouping is necessary in subtraction. Let us see an example to understand this. Subtract 2-digit numbers mentally with regrouping Example 16: Subtract mentally: 29 from 56 Solution: Follow these steps to subtract mentally. Steps Solved Solve this 29 from 56 46 from 83 83 = ___ + ____ Step 1: Regroup the two 29 = 20 + 9 46 = ___ + ____ given numbers as tens and 56 = 50 + 6 ones. Regroup the sum if it is equal to or more than 10. Step 2: Check if the ones 6 – 9 is not possible. So, ____ – ____ is possible (True/ can be subtracted. If not, regroup the tens. False). If it is true, subtract. If it regroup the tens. is false, regroup. Add 10 ones to 6 to get Add ten ones to ones and 16 and subtract 1 ten Add 10 ones to ___ to get reduce 1 ten from tens. from 5 tens to get 4 tens. ____ and subtract 1 ten from ____ tens to get ____ tens. Step 3: Subtract the ones of 16 – 9 = 7 ____ – ____ = ____ the two numbers mentally. Step 4: Subtract the tens of 4 tens – 2 tens = 2 tens ____ – ____ = ____ the two numbers mentally. ____ – ___ = ____ Step 5: Write the answers So, 56 – 29 = 27. from steps 3 and 4 together as the difference. Subtraction 55
Application We have seen that it is easy to subtract two 2-digit numbers mentally. In some real-life situations, we use mental subtraction of numbers. Let us see a few examples. Example 17: Manoj has 64 notebooks. He sold 45 notebooks. How many notebooks are left with him? Solve mentally. Solution: Number of notebooks Manoj has = 64 Number of notebooks he sold = 45 The number of notebooks remaining with him = 64 – 45 = 19 Therefore, Manoj has 19 notebooks left with him. Example 18: 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 the Term II than in the Term I? Solve mentally. Solution: Number of days Alisha went to school in Term I = 49 Number of days she went to school in Term II = 65 Difference in number of days = 65 – 49 = 16 Therefore, Alisha went to school 16 days more in Term II than in Term I. Higher Order Thinking Skills (H.O.T.S.) We have seen mental subtraction of two 2-digit numbers. Let us now see a real-life example where we might have to add and subtract numbers mentally. Example 19: Renu had ` 80. She bought guavas for ` 25 and bananas for ` 17. Calculate mentally the money that Renu has to pay the fruit seller. Also calculate mentally the money left with her. Solution: Total money Renu had = ` 80 Money she spent on guavas = ` 25 Money she spent on bananas = ` 17 To find the money she has to give the fruit seller, Renu has to add the prices of guavas and bananas. That is, ` 25 + ` 17 = ` 42. 56
To find the money remaining with her, Renu has to subtract this sum from the total money she had. So, ` 80 – ` 42 = ` 38. Therefore, ` 38 is left with Renu. Drill Time Concept 5.1: Estimate the Difference between Two Numbers 1) Estimate these differences: a) 65 – 15 b) 48 – 16 c) 67 – 32 d) 896 – 432 e) 679 – 387 2) Word problems a) In a class, there are 562 students. Of them, 118 are from the red group, 321 are from the green group, and the rest are from the blue group. How many students are in the blue group? b) Sneha has 77 balloons. She gives 42 balloons to her sister. About how many balloons remain with Sneha? Concept 5.2: Subtract 3-digit and 4-digit Numbers 3) Subtract 3-digit numbers with regrouping. a) 254 – 173 b) 678 – 619 c) 147 – 129 d) 781 – 682 e) 356 – 177 4) Subtract 4-digit numbers without regrouping. a) 2341 – 1230 b) 7632 – 5120 c) 9763 – 2311 d) 7629 – 1318 e) 7589 – 1268 5) Subtract 4-digit numbers with regrouping. a) 7632 – 1843 b) 4391 – 2482 c) 9843 – 7943 d) 8325 – 5436 e) 6893 – 3940 Subtraction 57
6) Word problems a) A stick is 8745 cm long. A 4313 cm long piece is cut from it. What part of the stick is remaining? b) R aj stays 5786 m away from Matin’s house. Raj travelled 3825 m of the distance. What is the distance yet to be covered by Raj to reach Matin’s house? Concept 5.3: Subtract 2-digit Numbers Mentally 7) 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 8) 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 9) 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? 58
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) Representing 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 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? Solution: Number of students in one class = 54 students H TO Number of classes in the school = 8 3 Number of students in the school = 54 × 8 54 Therefore, the total number of students in the school = 432 ×8 432 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? 62
Solution: The distance that Manoj travelled in a day = 7 km H TO He travels the same distance every day. The distance he 3 travels in 25 days = 25 × 7. 25 Therefore, Manoj travels 175 km in 25 days. ×7 175 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? 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 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 at the end of the year? Solution: Amount saved by Payal per month = ` 175 Th H T O Number of months in a year = 12 11 Total money saved in a year = 175 × 12 175 Therefore, Payal has ` 2100 at the end of the year. × 12 11 350 1750 2100 Multiplication 67
Example 10: Rohan ran 315 m every day for a week. How many metres did he run in that week? Th H T O Solution: 1 week = 7 days 13 Distance run by Rohan in a day = 315 m Distance he ran in a week = 315 m × 7 = 2205 m 315 ×7 So, Rohan covered a total distance of 2205 m in one 2205 week. 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 proTdruacitnis 3M6 ayndBrsuaminis 13. 36 = 1 × 36; 36 = 2 × 18; 36 = 3 × 12; 36 = 4 × 9 and 36 = 6 × 6. Of 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. 68
Concept 6.3: Double 2-digit and 3-digit Numbers Mentally Think Farida has 23 red beads. Her friend has double the number of beads. Farida wants to know the number of beads her friend has. Do you know how to find that mentally? Recall We have learnt mental addition and subtraction in the previous chapters. Let us now learn to double a given number mentally. To double a number, we must be thorough with the multiplication table of 2. For example, 5 × 2 = 10, 3 × 2 = 6, 10 × 2 = 20 and so on. & Remembering and Understanding Doubling a number means multiplying by 2. Let us now understand to double a 2-digit number mentally through a few examples. Example 13: Double the number 53. Solution: To double the given number, follow these steps. Steps Solved 53 Solve this 41 Step 1: Multiply the tens digit by 2. The tens digit is 5. The tens digit is ____. So, 5 × 2 = 10. So, ___ × 2 = ___. The ones digit is ___ Step 2: If the ones digit is less than or The ones digit is 3. ___ < ___ (True/ False) equal to 4, write the product in step 3 < 4 (True) 1 as it is. If not, add 1 to it and write. Step 3: Multiply the ones digit by 2. 3 × 2 = 6 ___ × 2 = ___ Step 4: Write the products in steps 53 × 2 = 106 ___ × 2 = ___ 1 and 3 together. This gives us the double of the given number. Multiplication 69
Application We have learnt to double 2-digit numbers mentally. Let us now see a few examples where we apply this concept. Example 14: Rohit has 14 shirts. His brother has double the number of shirts than he has. How many shirts does Rohit’s brother have? Solution: Number of shirts Rohit has = 14 Number of shirts Rohit’s brother has = Double the number of shirts that Rohit has = 14 × 2 = 28 Therefore, Rohit’s brother has 28 shirts. Example 15: Sony is 36 years old. Her aunt’s age is double the age of Sony. How old is Sony’s aunt? Solution: Sony’s age = 36 years Age of Sony’s aunt = Double that of Sony’s age = 36 years × 2 = 72 years Therefore, Sony’s aunt is 72 years old. Higher Order Thinking Skills (H.O.T.S.) Doubling a 3-digit number is similar to doubling a 2-digit number. Let us now see some examples. Example 16: Double the number 125. Solution: To double the given number, follow these steps. Steps Solved Solve this 125 293 Step 1: Multiply the number formed by the two leftmost digits by 2. The number formed by The number formed by the two leftmost digits the two leftmost digits is is 12. 12 × 2 = 24. ____. So, ___ × 2 = ___. 70
Steps Solved Solve this 125 293 Step 2: If the ones digit of the given number is less than or equal to 4. The ones digit is 5. The ones digit is __ write the product in step 1 as it is. If ___ < ___ (True/ False) not, add 1 to it and write. 5 < 4 (False) ___ × 2 = ___. 24 + 1 = 25 Its ones digit is ___. Step 3: Multiply the ones digit by 2. 5 × 2 = 10 So, ___ × 2 = ___. Its ones digit is 0. Step 4: Write the products in steps So, 125 × 2 = 250 1 and 3 together. This gives the double of the given number. 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) There were 23 students in one group. The school had 4 such groups. How many students were there in all the groups? b) Viraj 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 4) 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 5) 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 Multiplication 71
6) Multiply 3-digit numbers by 2-digit numbers. a) 769 × 21 b) 759 × 10 c) 578 × 42 d) 619 × 66 e) 290 × 30 7) 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? Concept 6.3: Double 2-digit and 3-digit Numbers Mentally 8) Double the given numbers mentally. a) 23 b) 52 c) 61 d) 10 e) 74 9) Word problems a) Rohan bought 42 books in Year I and double the number in Year II. How many books did he buy in Year II? b) Sonal earned ` 28 on Monday. She earned double the amount on Tuesday. How much did she earn on Tuesday? 72
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