Steps Solved Solve these Step 2: Add the tens and T Th Th H T O also the carry forward (if T Th Th H T O any) from the previous step. 11 2 5347 Write the sum under the tens. 4 8415 Regroup if needed. + 2 0098 + 6 2567 Step 3: Add the hundreds 13 T Th Th H T O and also the carry forward 1 7298 (if any) from the previous T Th Th H T O step. Write the sum under the 11 + 2 6543 hundreds place. Regroup if needed. 4 8415 + 2 0098 Step 4: Add the thousands and also the carry forward 513 (if any) from the previous step. Write the sum under the T Th Th H T O thousands place. Regroup if 11 needed. 4 8415 + 2 0098 8513 Step 5: Add the ten T Th Th H T O T Th Th H T O thousands and also the carry 11 forward (if any) from the 3 4765 previous step. Write the sum 4 8 4 15 + 2 1178 under the ten thousands + 2 0 0 98 place. 6 8 5 13 Therefore, 48415 + 20098 = 68513. Subtract 5-digit numbers Example 4: Subtract: 56718 – 16754 Solution: Arrange the numbers in columns, one below the other. 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 5 6718 9 7 0 54 under the ones place. − 1 6754 − 2 3 5 67 4 Addition and Subtraction 47
Steps Solved Solve these T Th Th H T O Step 2: Subtract the tens. T Th Th H T O That is, 1 − 5, which is not 7 5400 possible. 6⁄ 1⁄1 − 3 2689 Regroup the hundreds to 5 6718 T Th Th H T O tens, subtract and write the − 1 6754 difference under the tens 5 4635 place. 64 − 1 2789 Step 3: Subtract the hundreds. That is, 6 − 7, T Th Th H T O T Th Th H T O which is not possible. 16 8 9576 Regroup the thousands to 6⁄⁄ − 4 5689 hundreds, subtract and write 5 5 11 8 the difference under the 7 hundreds place. 6⁄ 1⁄ − 1 6754 964 Step 4: Subtract the T Th Th H T O thousands. That is, 5 − 6, which is not possible. 1⁄5 1⁄6 4⁄ 5⁄ 6⁄ 1⁄1 Regroup the ten thousands to thousands, subtract and 5 6 7 18 write the difference under − 1 6 7 54 the thousands place. Step 5: Subtract the ten 9 9 64 thousands, and write the difference under the ten T Th Th H T O thousands place. 4⁄ 15⁄⁄5 16⁄⁄6 1⁄1 Therefore, 56718 – 16754 = 39964. 5 6 7 18 − 1 6 7 54 3 9 9 64 ? Train My Brain Solve the following: a) 3456 + 2709 b) 42361 + 18194 c) 97972 – 10402 48
I Apply Addition and subtraction of 4-digit and 5-digit numbers are useful in our daily life. Here are a few examples. Example 5: Raju had ` 90005 with him. He bought clothes for ` 35289. How much money was left with him? Solution: Amount Raju had = ` 90005 T Th Th H T O Amount Raju spent on buying clothes = ` 35289 Amount left with him = ` 90005 – ` 35289 8⁄ 9⁄ 9⁄ 9⁄ 1⁄5 Therefore, the amount left with Raju is ` 54716. 90005 −35289 54716 Example 6: Preeti drove her car for 26349 km in six weeks and 38614 km in the next eight weeks. How many kilometres in all did she drive in 14 weeks? Solution: Distance Preeti drove in the first six weeks = T Th Th H T O 26349 km 11 2 6349 Distance she drove in the next eight weeks = 38614 km + 3 8614 6 4963 The total distance that Preeti drove = 26349 km + 38614 km Therefore, Preeti drove a total distance of 64963 km in 14 weeks. Example 7: Mohan’s uncle stays 8630 m away from Mohan’s house. Mohan travelled 6212 m of the distance. What is the distance yet to be covered by Mohan to reach his uncle’s house? Solution: Distance between Mohan’s house and his uncle’s house = 8630 m Distance travelled by Mohan = 6212 m Th H T O Remaining distance that Mohan has to travel 2 10 ⁄⁄ 8630 = 8630 m – 6212 m −6212 Therefore, Mohan has to travel 2418 m more to 2418 reach his uncle’s house. Addition and Subtraction 49
I Explore (H.O.T.S.) We can frame word problems on addition and subtraction. Example 8: Payal and Suma have 1284 and 5215 stamps respectively. Frame an addition problem. Solution: An addition problem contains words such as - in all, total, altogether and so on. So, the question can be “Payal and Suma have 1284 and 5215 stamps respectively. How many stamps do they have altogether?” Example 9: Frame a word problem based on the given subtraction. 50000 – 49100 = 900 Solution: A subtraction problem contains words such as - difference, left, remaining and so on. So, the question can be “Ramesh had ` 50000. He gave ` 49100 to his brother. Find the amount left with Ramesh”. Maths Munchies Always remember that when we add a number to itself, the sum is double the original number. If we subtract a number from itself, the difference is zero. For example, 2000 + 2000 = 4000 (which is double of 2000) and 2000 – 2000 = 0. Connect the Dots Social Studies Fun In 1557, Robert Recorde shortened “is equal to” to two long, parallel lines. This gave the presently used ‘equal to’ sign. He used this to avoid repeating himself 200 times in his book. English Fun ‘Addition’ is a noun. Write the verb for ‘addition’. Can you say what the adjective form for this word is? Share with your friends. 50
Drill Time 4.1 Add and Subtract 4-digit and 5-digit Numbers 1 ) Add the following: a) 5624 + 1218 b) 42584 + 23568 c) 4721 + 1311 d) 65312 + 25842 e) 35216 + 42355 2) Subtract the following: a) 5943 – 1256 b) 86531 – 65372 c) 95361 – 46472 d) 11213 – 11206 e) 34536 – 15623 3) Word problems a) Seeta went to purchase a TV from an electronics shop. The price of the TV was ` 25689. She paid the shopkeeper ` 50000. How much money will she receive back? b) Rohan collected 1256 beads for a design. Sohan collected 2563 beads for the same design. How many beads did they collect in all? A Note to Parent Play this fun game with your child. Shuffle a deck of cards. Draw a card randomly from it. Multiply the number on the card by 100. The number obtained is your score. Note it down on a piece of paper. All those playing the game should do the same. Continue the game for a few rounds or till all the cards are drawn from the deck. Add the score obtained by each player. The one with the highest score wins. Addition and Subtraction 51
5Chapter Multiplication I Will Learn About • multiplication of 2-digit and 3-digit numbers. • multiplication using lattice algo- rithm. • mental multiplication. 5.1 Multiplication of 2-digit Numbers and 3-digit Numbers I Think Surbhi went to a stadium to watch a football 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? I Recall We have learnt to multiply 2-digit numbers by 1-digit numbers. 52
Let us solve the following to revise the concept of multiplication. a) T O b) H T O c) H T O d) H T O e) H T O 39 56 89 75 90 ×2 ×3 ×4 ×5 ×4 I Remember and Understand We have learnt the multiplication tables from 2 to 10. Let us now learn the multiplication tables from 11 to 20. Multiplication Tables 11 12 13 14 15 11 × 1 = 11 12 × 1 = 12 13 × 1 = 13 14 × 1 = 14 15 × 1 = 15 11 × 2 = 22 12 × 2 = 24 13 × 2 = 26 14 × 2 = 28 15 × 2 = 30 11 × 3 = 33 12 × 3 = 36 13 × 3 = 39 14 × 3 = 42 15 × 3 = 45 11 × 4 = 44 12 × 4 = 48 13 × 4 = 52 14 × 4 = 56 15 × 4 = 60 11 × 5 = 55 12 × 5 = 60 13 × 5 = 65 14 × 5 = 70 15 × 5 = 75 11 × 6 = 66 12 × 6 = 72 13 × 6 = 78 14 × 6 = 84 15 × 6 = 90 11 × 7 = 77 12 × 7 = 84 13 × 7 = 91 14 × 7 = 98 15 × 7 = 105 11 × 8 = 88 12 × 8 = 96 13 × 8 = 104 14 × 8 = 112 15 × 8 = 120 11 × 9 = 99 12 × 9 = 108 13 × 9 = 117 14 × 9 = 126 15 × 9 = 135 11 × 10 = 110 12 × 10 = 120 13 × 10 = 130 14 × 10 = 140 15 × 10 = 150 16 17 18 19 20 16 × 1 = 16 17 × 1 = 17 18 × 1 = 18 19 × 1 = 19 20 × 1 = 20 16 × 2 = 32 17 × 2 = 34 18 × 2 = 36 19 × 2 = 38 20 × 2 = 40 16 × 3 = 48 17 × 3 = 51 18 × 3 = 54 19 × 3 = 57 20 × 3 = 60 16 × 4 = 64 17 × 4 = 68 18 × 4 = 72 19 × 4 = 76 20 × 4 = 80 16 × 5 = 80 17 × 5 = 85 18 × 5 = 90 19 × 5 = 95 20 × 5 = 100 16 × 6 = 96 17 × 6 = 102 18 × 6 = 108 19 × 6 = 114 20 × 6 = 120 16 × 7 = 112 17 × 7 = 119 18 × 7 = 126 19 × 7 = 133 20 × 7 = 140 16 × 8 = 128 17 × 8 = 136 18 × 8 = 144 19 × 8 = 152 20 × 8 = 160 16 × 9 = 144 17 × 9 = 153 18 × 9 = 162 19 × 9 = 171 20 × 9 = 180 16 × 10 = 160 17 × 10 = 170 18 × 10 = 180 19 × 10 = 190 20 × 10 = 200 Multiplication 53
Let us now learn to multiply: Standard algorithm is the 1) 3-digit numbers by 1-digit numbers. method of multiplication 2) 2-digit numbers by 2-digit numbers. in which the product is 3) 3-digit numbers by 2-digit numbers. regrouped as ones and tens. Multiply a 3-digit number by a 1-digit number 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 the following example. Example 1: Multiply: 513 × 5 Solution: Follow these steps to multiply the given numbers. Steps Solved Solve these H TO Step 1: Multiply the ones. Regroup if the HT O product has two digits. Carry the tens digit 1 635 of the product to the tens place and write 3 ×7 its ones digit under the ones place. 51 5 × 5 Step 2: Multiply the tens. Add the carry H TO H TO over (if any) to the product. Regroup if the 1 product has two digits. Carry the tens digit 444 of the product to the hundreds place and 513 ×8 write its ones digit under the tens place. ×5 65 Step 3: Multiply the hundreds. Add the Th H T O H TO carry over (if any) to the product. Regroup 1 if the product has two digits. Carry the 342 tens digit of the product to the thousands 513 ×5 place. Write its ones digit under the ×5 hundreds place and the tens digit under the thousands place. 2565 Multiply a 2-digit number by a 2-digit number Let us multiply 2-digit numbers by 2-digit numbers through a step-by-step procedure. Consider the following example. 54
Example 2: Multiply: 24 × 34 Solution: Follow these steps to multiply the given numbers. Steps Solved Solve these Step 1: Multiply the ones by the ones digit H TO of the multiplier. 4 × 4 = 16 TO 41 1 ×22 Write 6 in the ones place of the product. 24 Write 1 in the tens place as carry over. ×34 H TO 52 Step 2: Multiply the tens digit by the ones 6 digit of the multiplier. 2 × 4 = 8 ×23 H TO Add the carry over from the previous step 1 H TO to the product. So, 8 + 1 = 9. Write 9 in the 24 63 tens place of the product. ×34 ×23 Step 3: Write 0 in the ones place under 96 the product obtained from the previous steps. Now multiply the ones digit by the H TO tens digit of the multiplier. 1 3 × 4 = 12 1 Write 2 in the tens place, below 9 and 24 carry the tens digit,1, to the tens place. ×34 Step 4: Multiply the tens by the tens digit 96 of the multiplier. 20 3×2=6 H TO 1 Add the carry over from the previous 1 step. So, 6 + 1 = 7. Write 7 in the hundreds 24 place. ×34 Step 5: Add the products and write the 96 sum, which is the required product. 720 Th H T O 1 1 24 ×34 196 +720 816 Multiplication 55
Multiply a 3-digit number by a 2-digit number Multiplication of 3-digit numbers by 2-digit numbers is similar to multiplication of two 2-digit numbers. It may sometimes involve regrouping too. Let us understand this concept through a step-by-step procedure. Consider the following example. Example 3: Multiply: 243 × 34 Solution: Arrange the numbers in columns, as shown. Steps Solved Solve these Step 1: Multiply the ones by the ones digit H TO of the multiplier. 3 × 4 = 12 H TO 1 453 Write 2 in the ones place of the product. ×13 Carry the 1 to the tens place. 243 ×34 Step 2: Multiply the tens digit by the ones digit of the multiplier. 4 × 4 = 16 2 Add the carry over from the previous H TO step. So, 16 + 1 = 17. Write 7 in the tens place of the product and 1 in the 11 hundreds place as carry over. 243 ×34 72 Step 3: Multiply the hundreds by the ones H TO H TO digit of the multiplier. 2 × 4 = 8 11 263 Add the carry over from the previous 243 ×23 step. So, 8 + 1 = 9. Write 9 in the hundreds ×34 place of the product. 972 Step 4: Write 0 in the ones place below HTO the product obtained from the previous 11 steps. 243 ×34 Multiply the ones by the tens digit of the 972 multiplier. Write the product under the tens place. 90 3×3=9 Write 9 below 7 in the tens place of the previous product. 56
Steps Solved Solve these H TO Step 5: Multiply the tens by the tens digit H TO 141 of the multiplier. ×22 1 4 × 3 = 12 H TO 11 352 Write 2 in hundreds place of the 243 ×23 product and 1 in hundreds place of the ×34 multiplicand as carry over. 972 290 Step 6: Multiply the hundreds by the tens Th H T O digit of the multiplier. 2×3=6 1 Add the carry over from the previous 11 step. So, 6 + 1 = 7. Write 7 in the thousands 243 place of the product. ×34 972 7290 Step 7: Add the products and write the Th H TO sum. Regroup if necessary. The sum is the 1 required product. 11 243 ×34 11 972 +7 2 9 0 8 2 6 2 ? Train My Brain Solve the following: a) 222 × 8 b) 92 × 32 c) 632 × 22 Multiplication 57
I Apply Let us solve a few real-life examples involving multiplication of 3-digit numbers. Example 4: There are 18 desks in a classroom. There are 11 such classrooms in the school. There are 13 such schools in the city. How many desks are there in total in all the schools? Th H T O Solution: Number of desks in a classroom = 18 22 Number of classrooms = 11 198 ×13 Number of desks in 11 classrooms = 18 × 11 = 198 11 Number of schools = 13 594 Number of desks in 13 schools = 198 × 13 +1980 2574 Therefore, there are 2574 desks in total in all the schools. Example 5: 354 students went to school from each of the two neighbouring towns. How many students in all went to school? Solution: Number of students from each town = 354 H TO Number of towns = 2 1 Total number of students who went to school = 354 × 2 3 54 ×2 Therefore, 708 students went to the school from both the towns. 7 0 8 Consider the following to multiply a 3-digit number by a 3-digit number. Multiply: 159 × 342 T Th Th H T O 12 23 11 159 ×342 1 1 3 1 8 → Multiply the multiplicand by the ones of the multiplier. + 159 × 2 ones. +4 6 3 6 0 → Multiply the multiplicand by the tens of the multiplier. 159 × 4 tens. 7 7 0 0 → Multiply the multiplicand by the hundreds of the multiplier. 159 × 3 hundreds. 5 4 3 7 8 → Add the products and write the sum. Therefore, 159 × 342 = 54378. 58
I Explore (H.O.T.S.) We can frame word problems based on the multiplication of 2-digit numbers and 3-digit numbers. Example 6: Frame a word problem using 48 × 98. Solution: Using 48 × 98, we can frame a word problem as \"There are 48 chalk pieces in each box. If there are 98 such boxes, find the total number of chalk pieces\". Example 7: Frame a word problem using 792 × 19. Solution: U sing 792 × 19, we can frame a word problem as \"There are 792 candies in a big container. How many candies will 19 such containers have?\" 5.2 Multiply Using Lattice Algorithm I Think Surbhi and her friends visited her aunt's plant nursery. There she saw an arrangement with equal number of pots in rows. She wanted to know the total number of pots using a simple method. Do you know any such method? I Recall We know multiplication of 2-digit numbers by 1-digit numbers using lattice algorithm. Let us recall the concept by solving the following: 2 2× 4 1× 5 0× 4 3 4 Multiplication 59
I Remember and Understand We have learnt to multiply a 2-digit number by a 1-digit number using standard algorithm. Let us now learn to multiply a 2-digit number and a 3-digit number by a 2-digit number using a simpler method than the standard algorithm. This method is called the Lattice algorithm. Example 8: Multiply using lattice algorithm: a) 43 × 52 b) 168 × 48 Solution: To multiply numbers using the lattice algorithm, follow these steps: Steps Solved Solved a) 43 × 52 b) 168 × 48 Step 1: Construct a lattice as shown, such that: (a) Number of rows = Number of digits in the multiplier. (b) Number of columns = Number of digits in the multiplicand. Draw diagonals to divide each box into parts as shown. Step 2: Write the multiplicand along 4 3× 1 6 8× the top of the lattice and the 5 4 multiplier along the right, one digit for each row or column. 2 8 Step 3: Multiply each digit of the 4 3× 1 6 8× multiplicand by the tens digit of the 2015 5 multiplier. Write the products in the 0 2 3 4 cells where the corresponding rows 2 4 4 2 and columns meet. Write the tens digit of the product in the upper 8 half and its ones digit in the lower half as shown. 60
Steps Solved Solved a) 43 × 52 b) 168 × 48 Step 4: Multiply each digit of the multiplicand by the ones digit of 4 3× 16 8× the multiplier. Write the products 2015 5 in the cells as done in the previous 02 3 4 step. 44 2 If the product is a single digit 0 0 2 04 6 8 number, put 0 in the tens place. 8 6 88 4 (2 × 3 = 6) = 06 Step 5: Add the numbers along the 4 3 × 16 8 × diagonals from the right to find the 2 1 5 3 4 product. Regroup if needed. Write 20 02 the sum from left to right. 5 2 44 2 8 10 0 0 04 6 28 6 28 8 4 3 8 01 6 4 Therefore, 6 43 × 52 = 2236. Therefore, 168 × 48 = 8064. 6 3× Solve these × 171 × 3 172 3 4 3 1 Train My Brain 2 Therefore, 63 × 33 = _____. Therefore, 172 × 42 = _____. Therefore, 171 × 31 = _____. ? Train My Brain Multiply using the lattice algorithm method: a) 54 × 3 b) 78 × 21 c) 375 × 27 Multiplication 61
I Apply Let us now see a few real-life examples where we use the lattice algorithm for the 3-digit numbers. 34 5 × 1 Example 9: There are 345 students in each class. 0 00 Pooja’s school has 12 such classes. 2 How many students are there in her 34 5 Solution: school? 1 0 0 1 Number of students in each class = 345 6 8 0 4 Number of such classes in Pooja’s 11 4 0 school = 12 Total number of students = 345 × 12 Therefore, there are 4140 students in Pooja’s school. Example 10: 42 people were sitting in a row of a stadium to enjoy a cricket match. How many people would be there in all if 35 such rows in the stadium were filled? Solution: Number of people sitting in one row = 42 4 2× Number of such rows filled = 35 10 Total number of people in 35 rows 1 2 63 = 42 × 35 Therefore, there are 1470 people in the 2 1 5 stadium. 40 0 I Explore (H.O.T.S.) 70 We know how to multiply numbers using lattice algorithm. Let us see if we can analyse and solve the following. Example 11: Find the missing numbers in the given 2 3 ?× lattice multiplication. 0 1 n2 8 4 0 8 2 23___ × 4___= 9954 00 1 Solution: We see that the box at the top right 4? corner has the number 28. It is the 94 6 product of 4 and ?. 954 To find the missing number, divide 28 by 4. We get 7. 62
That is, 4 × ? = 28 4 × 7 = 28. So, 7 is the first unknown number. Similarly, the box in the bottom left corner has 04. It is the product of 2 and ?. To find the missing number, divide 4 by 2. We get 2. That is, 2 × ? = 04 2 × 2 = 04. So, the second unknown number is 2. Therefore, the required missing numbers are 7 and 2 so that 237 × 42 = 9954. 5.3 Mental Maths Techniques: Multiplication Let us now understand how to complete multiplication facts mentally. Steps Solved Solve this 14 × 6 12 × 8 Step 1: Check by how much The larger number is 14 The larger number is ____, the larger number is more than and it is 4 more than 10. and it is ______ more than 10. (If the larger of the given 10. numbers is more than 10.) Step 2: Write the larger 10 + 4 = 14 ______ + ______ = _______ number as the sum of 10 and the number obtained in the 10 × 6 = 60 and 10 × ________ = _________ previous step. 4 × 6 = 24 and _______ × ______ = _______ Step 3: Multiply the numbers of the sum separately by the smaller number given, using memorised tables of 1 to 10. Step 4: Add both the products 60 + 24 = 84 ______ + ______ = ______ Therefore, 12 × 8 = ______. from step 3 mentally. This gives Therefore, 14 × 6 = 84. the required product. ? Train My Brain Find the product in each of the following mentally. a) 18 × 7 b) 13 × 4 c) 11 × 6 d) 24 × 8 e) 76 × 5 Multiplication 63
Maths Munchies Let us see how to multiply two 2-digit numbers, 13 and 21, quickly. For the multiplicand 13, draw 1 tens as 1 green line 1 6 and 3 ones as 3 pink lines. For the multiplier 21, draw 1 ones of 21 3 2 tens as 2 red lines and 1 ones as 1 yellow line. 2 3 ones of 13 Count the number of dots at each intersection 1 tens of 13 2 tens as shown. Add the numbers in the middle column. of 21 Here, 6 + 1 = 7. Write the number starting from the left. So, the product is 273. This method is also called as the Chinese-stick multiplication method. Connect the Dots English Fun The word ‘Lattice’ in lattice algorithm is not an English word originally. The word is taken from ‘lattis’, in old French language which itself has been taken from ‘latte’ or ‘lath’, that means the wire mesh used for backing in the old German language. Social Studies Fun The oldest known multiplication table was found written on bamboo strips in China which are said to be 2300 years old. Modern multiplication tables are said to have been written down by the famous Greek mathematician Pythagoras. It is also called the Table of Pythagoras in many other languages. Drill Time 5.1 Multiplication of 2-digit Numbers and 3-digit Numbers 1) Multiply a 3-digit number by a 1-digit number. a) 247 × 7 b) 509 × 1 c) 892 ×9 d) 731 × 1 e) 435 × 4 64
2) Multiply a 2-digit number by a 2-digit number. a) 17 × 16 b) 39 × 24 c) 99 × 19 d) 43 × 32 e) 76 × 48 3) Multiply a 3-digit number by a 2-digit number. a) 161 × 23 b) 276 × 11 c) 157 × 38 d) 375 × 82 e) 198 × 22 4) Multiply a 3-digit number by a 3-digit number. a) 674 × 124 b) 345 × 238 c) 925 × 759 d) 555 × 725 e) 804 × 682 5) Word problems a) Pranav makes 25 cotton bags in a day. How many bags will he be able to make in the year 2017? (Hint: 2017 is not a leap year.) b) Tanya bought a few gifts of the same kind for Christmas for her 14 friends. If one gift costs ` 27, then how much money in all did she spend for the gifts? 5.2 Multiply Using Lattice Algorithm 6) Multiply a 2-digit number by a 2-digit number. a) 24 × 32 b) 56 × 15 c) 13 × 39 d) 67 × 51 e) 78 × 42 7) Multiply a 3-digit number by a 2-digit number. a) 158 × 17 b) 451 × 39 c) 651 × 67 d) 721 × 41 e) 329 × 78 8) Word problems a) A movie theatre sold 127 tickets for a movie. The cost of one ticket was ` 85. How much money did the theatre owner earn? b) There are 47 students in Class 3. Answer booklets were given to each student for a Maths exam. If one answer booklet has 15 pages, then how many sheets of paper in all were used for the exam? A Note to Parent Help your child understand and practise the Chinese-stick method of multiplication using straws of different colours. Multiplication 65
Division6Chapter I Will Learn About • dividing 2-digit and 3-digit numbers by 1-digit numbers. • solving real-life examples based on division. 6.1 Divide 3-digit Numbers by 1-digit Numbers I Think Surbhi and seven of her friends want to share 350 papers equally among themselves. Do you think she can divide the papers without some of them being left? I Recall In grade 3, we have learnt about equal grouping and repeated subtraction. We have also learnt about the relation between multiplication and division. Equal grouping: Having equal number of objects in each group into which a collection of objects is divided is called equal grouping. Repeated subtraction: Subtracting the same numbers over and over from a given number is called repeated subtraction. 66
Let us recall these concepts by answering the following questions. a) 15 – 5 = 10; _____ – 5 = 5; 5 – _____ = 0 b) 9 – _____ = 6; 6 – 3 = _____; ______ – ______ = 0 c) 7 × 5 = 35 is the multiplication fact of 35 ÷ 7 = _____ . d) 40 ÷ 8 = 5 is the division fact of 5 × _____ = 40. e) 28 ÷ 4 = __________ I Remember and Understand Dividing a number by 1 gives the We can make equal groups and divide vertically. Multiplication quotient same as tables can be used to divide numbers vertically. Let us see an the dividend. example. Divide a 2-digit number by a 1-digit number (1-digit quotient) Example 1: Divide: 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 the 5)45 )8 56 Dividend = _____ divisor as shown: Divisor = ______ 45 = 5 × 9 − Quotient = ____ )Divisor Dividend 9 Remainder = _____ Step 2: Find the multiplication fact )5 45 in the table of the divisor which has the dividend as the product. − 45 Step 3: Write the factor other than the divisor as the quotient. Write the product in the multiplication fact below the dividend. Step 4: Subtract the product 9 )4 36 Dividend = _____ from the dividend and write the Divisor = ______ difference below the product. )5 45 − Quotient = ____ Remainder = _____ This difference is called the − 45 remainder. 00 45 = Dividend 5 = Divisor 9 = Quotient 0 = Remainder Division 67
Note: If the remainder is zero, the divisor is said to divide the dividend exactly. Checking for the correctness of division The multiplication fact of the division is used to check its correctness. Step 1: Compare the remainder and the divisor. The remainder must always be less than the divisor. Step 2: Check if (Quotient × Divisor) + Remainder = Dividend. If this is true, the division is correct. Let us now check if the division in example 1 is correct or not. Step 1: Remainder < Divisor 0 < 5 (True) Step 2: Quotient × Divisor 9 × 5 = 45 Step 3: (Quotient × Divisor) + Remainder 45 + 0 = 45 Step 4: (Quotient × Divisor) + Remainder = Dividend 45 = 45 (True) Therefore, the division is correct. Note: a) The division is incomplete if Remainder > Divisor (OR) Remainder = Divisor. b) The division is incorrect if (Quotient × Divisor) + Remainder ≠ Dividend. 2-digit quotients 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 an example. Example 2: Divide: 57 ÷ 3 Solution: Follow these steps to divide a 2-digit number by a 1-digit number. Steps Solved Solve these 5>3 Step 1: Check if the tens digit of the )5 60 dividend is greater than the divisor. 1 − Step 2: Divide the tens and write the )3 57 quotient. − −3 Write the product of quotient and divisor, Dividend = _____ below the tens digit of the dividend. 1 Divisor = ______ Quotient = ____ Step 3: Subtract and write the difference )3 57 Remainder = ___ (= Remainder). −3 2 68
Steps Solved Solve these Step 4: Check if difference < divisor is true. 2 < 3 (True) )3 42 Step 5: Bring down the ones digit of the 19 dividend and write it beside the remainder. − )3 57 − Step 6: Find the largest number in the − 3↓ multiplication table of the divisor that can 27 Dividend = _____ be subtracted from the 2-digit number in Divisor = ______ the previous step. 3 × 8 = 24 Quotient = ____ Remainder = ___ 19 )3 57 − 3↓ 27 3 × 9 = 27 3 × 10 = 30 24 < 27 < 30. So, 27 is the required number. Step 7: Write the factor of the required 19 )3 39 number, other than the divisor, as the quotient. Write the product of the divisor )3 57 − and the quotient below the 2-digit number. Subtract and write the difference. − 3↓ − 27 Step 8: If remainder < divisor is true, stop the Dividend = _____ division. − 27 Divisor = ______ If the remainder is equal to or greater than 00 Quotient = ____ the divisor, the division is incorrect. So, Remainder = ___ check the division and correct it to get a 0 < 3 (True) remainder less than the divisor. Step 9: Write the quotient and the Quotient = 19 remainder. Remainder = 0 Step 10: Check if (Divisor × Quotient) + 3 × 19 = 57 Remainder = Dividend is true. 57 + 0 = 57 (If not, then the division is incorrect.) 57 = 57 (True) Division 69
Divide a 3-digit number by a 1-digit number (2-digit quotients) 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 an example. Example 3: Divide: 265 ÷ 5 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 5)265 )4 244 of the dividend is greater than the divisor. If it is not, consider its tens 2 is not greater than 5. − digit also. So, consider 26. − Step 2: Find the largest number in 5 the table of the divisor that can be Dividend = _____ subtracted from the 2-digit number )5 265 Divisor = ______ of the dividend. Write the quotient. Quotient = ____ Write the product of the quotient and − 25 Remainder = ___ the divisor below the dividend. 5 × 4 = 20 )6 366 5 × 5 = 25 5 × 6 = 30 − 25 < 26 − Step 3: Subtract and write the 5 remainder. )5 265 − 25 1 Step 4: Check if remainder < divisor 1 < 5 (True) is true. (If not, then the division is incomplete.) 5 Step 5: Bring down the ones digit )5 265 of the dividend. Write it beside the remainder. − 25↓ 15 Step 6: Find the largest number in the 5 Dividend = _____ multiplication table of the divisor that Divisor = ______ can be subtracted from the 2-digit )5 265 Quotient = ____ number in the previous step. Remainder = ___ − 25↓ 15 5 × 2 = 10 5 × 3 = 15 5 × 4 = 20 15 is the required number. 70
Steps Solved Solve these 53 Step 7: Write the factor other than )9 378 the divisor, as the quotient. Write )5 265 the product of the divisor and the − quotient below the 2-digit number. − 25↓ Then, subtract them. 15 − Step 8: Check if remainder < divisor is − 15 Dividend = _____ true. Stop the division. (If not, then the 00 Divisor = ______ division is incomplete.) Quotient = ____ Step 9: Write the quotient and the 0 < 5 (True) Remainder = ___ remainder. Step 10: Check if (Divisor × Quotient) Quotient = 53 + Remainder = Dividend is true. (If Remainder = 0 not, then the division is incorrect.) 5 × 53 = 265 265 + 0 = 265 265 = 265 (True) 3-digit quotients Example 4: Divide: 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 divisor. )7 784 )8 984 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. Step 3: Subtract and write the remainder. −7 1 )7 784 −7 0 Step 4: Check if remainder < divisor is true. 0 < 7 (True) Dividend = _____ Divisor = ______ Step 5: Bring down the next digit of the 1 Quotient = ____ dividend. Check if it is greater than or equal Remainder = ___ to the divisor. )7 7 84 −7↓ 08 8>7 Division 71
Steps Solved Solve these Step 6: Find the largest number in the 11 )5 965 multiplication table of the divisor that can be subtracted from the 2-digit number in the )7 784 − previous step. Write the factor other than the divisor as the − 7↓ − quotient. 08 Write the product of the quotient and the − divisor below it. −7 Dividend = _____ 7×1=7<8 Divisor = ______ The required Quotient = ____ number is 7. Remainder = ___ Step 7: Subtract and write the remainder. 11 )2 246 Bring down the next digit (ones digit) of the dividend. )7 784 − Check if the dividend is greater than or equal to the divisor. If so, then continue with the − 7↓ − division. 08 − Step 8: Find the largest number in the −7 multiplication table of the divisor that can 14 Dividend = _____ be subtracted from the 2-digit number in the Divisor = ______ previous step. 14 > 7 Quotient = ____ Write the factor other than the divisor as the Remainder = ___ quotient. 112 Write the product of the quotient and the divisor below it. )7 784 − 7↓ 08 −7 14 − 14 Step 9: Subtract and write the remainder. 7 × 2 = 14 Check if it is less than the divisor. Stop the The required division. number is 14. 112 )7 784 − 7↓ 08 −7 14 − 14 00 72
Steps Solved Solve these Step 10: Write the quotient and the Quotient = 112 remainder. Remainder = 0 Step 11: Check if (Divisor × Quotient) + 7 × 112 = 784 Remainder = Dividend is true. (If not, then the division is incorrect.) 784 + 0 = 784 784 = 784 (True) ? Train My Brain Solve the following: a) 12 ÷ 4 b) 648 ÷ 8 c) 744 ÷ 4 I Apply Division of 2-digit numbers and 3-digit numbers is used in many real-life situations. Let us consider a few examples. Example 5: A school has 634 students, who are equally grouped into 4 houses. How many students are there in a house? Are there any students who are not grouped into any house? Solution: Number of students = 634 158 Number of houses = 4 )4 634 Number of students in a house = 634 ÷ 4. − 4↓ 23 Therefore, the number of students in each house = 158 − 20 The remainder in the division is 2. 34 − 32 Therefore, 2 students are not grouped into any house. 02 Example 6: A football game had 99 spectators. If each row in the stadium has only Solution: 9 seats, how many rows did the spectators occupy? 11 Number of spectators = 99 )9 99 Number of seats in each row = 9 − 9↓ Number of rows occupied by the spectators = 99 ÷ 9. 09 Therefore, 11 rows were occupied by the spectators. −9 0 Division 73
I Explore (H.O.T.S.) In all the divisions we have seen so far, we did not have a 0 (zero) in the dividend or the quotient. When a dividend has a zero, we place a 0 in the quotient in the corresponding place. Then, get the next digit of the dividend down and continue the division. Let us now understand the division of numbers that have a 0 (zero) in the dividend or quotient, through this example. Example 7: Divide: 505 ÷ 5 Solution: Solved Solve this 505 ÷ 5 804 ÷ 4 101 )4 804 )5 505 − − 5↓ − 00 − − 00 05 − 05 00 Maths Munchies Dividing a 2-digit number or a 3-digit number by 10: Observe the pattern in the division of the following examples: a) 562 ÷ 10 = Q 56 and R 2 d) 45 ÷ 10 = Q 4 and R 5 b) 325 ÷ 10 = Q 32 and R 5 e) 72 ÷ 10 = Q 7 and R 2 c) 687 ÷ 10 = Q 68 and R 7 We observe that the ones digit of the dividend is the remainder and the number formed by its remaining digits is the quotient. This helps us to do the divisions quickly. 74
Connect the Dots Science Fun There are different kinds of living things other than people on the Earth. All living things are divided into various groups. These divisions help scientists study various aspects about them. Social Studies Fun India is a very large country. It becomes difficult to take care of the citizens of the entire country by a small group of leaders. So, it is divided into 29 states and 7 union territories. Drill Time 6.1 Divide 3-digit Numbers by 1-digit Numbers 1) Divide 2-digit numbers by 1-digit numbers (1-digit quotient). a) 12 ÷ 2 b) 24 ÷ 6 c) 36 ÷ 6 d) 40 ÷ 8 e) 10 ÷ 5 2) Divide 2-digit numbers by 1-digit numbers (2-digit quotient). a) 12 ÷ 1 b) 99 ÷ 3 c) 48 ÷ 2 d) 65÷ 5 e) 52 ÷ 4 3) Divide 3-digit numbers by 1-digit numbers (2-digit quotient). a) 123 ÷ 3 b) 102 ÷ 2 c) 497 ÷ 7 d) 111 ÷ 3 e) 256 ÷ 4 4) Divide 3-digit numbers by 1-digit numbers (3-digit quotient). a) 456 ÷ 2 b) 112 ÷ 1 c) 306 ÷ 3 d) 448 ÷ 4 e) 555 ÷ 5 5) Word problems a) There are 260 chocolates in a jar that have to be distributed equally among 4 students. How many chocolates will each student get? b) There are 24 passengers in a bus. 2 passengers can sit on each seat. How many seats in the bus are occupied? Division 75
A Note to Parent Collect electricity bills of the past three months and show them to your child. Highlight the total amount and the cost per unit of electricity. Explain how the number of units of electricity utilised is calculated. Now help your child to calculate the number of units of electricity that the family has used in the past three months. 76
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