HINDI - FL - CLASS 7 - TS

Contents 1. Large Numbers ........................ 52 2. Addition .................................... 62 3. Subtraction ............................... 66 4. Multiplication ............................. 70 5. Division..................................... 78 6. HCF and LCM .......................... 85 7. Symmetry and Patterns............ 95 8. Four Operations ..................... 106 Class 5 Term 1

Large Numbers Indian place value system. Crores Lakhs Thousands Hundreds Tens Ones 100 Cr. 10 Cr. Cr. 10 Lakhs Lakhs Thousands Thousands 10 2 3 1 2 7 8 4 3 4 7 4 3 1 2 1 4 8 6 1 2 3 4 2 4 7 4 8 3 5 2 3 1 8 1 Writing the numbers in figures using comma 23,12,784: Twentythree lakhs, twelve thousand, seven hundred and eightyfour. 1,34,74,312: One crore, thirtyfour lakhs, seventyfour thousand, three hundred and twelve. 42, 48, 61, 234: Fortytwo crores, fortyeight lakhs, sixtyone thousand, two hundred and thirtyfour. 1,81,74,83,523: One hundred eightyone crores, seventyfour lakhs, eightythree thousand, five hundred and twentythree. 52

International place value system. Billions Millions Thousands Hundreds Tens Ones 100 B 10 B B 100 M 10 M 100 Th 10 Th 2 3 2 7 8 4 3 4 4 3 1 2 4 4 8 1 2 3 4 8 7 4 3 5 2 8 1 M Th 1 2 1 1 7 6 8 1. Two million, three hundred twelve thousand, seven hundred and eightyfour – 2,312,784. 2. Thirteen million, four hundred seventyfour thousand, three hundred and twelve - 13,474,312 3. Four hundred twentyfour million, eight hundred sixtyone thousand, two hundred and thirtyfour - 424,861,234 4. One billion, eight hundred seventeen million, four hundred eighty- three thousand, five hundred and twentyeight - 1,817,483,528 100 Cr. 10 Cr. Cr. 10 Lakhs Lakhs 10 Th Th Hundeds Tens Ones 10 Th Th Hundeds Tens Ones 100 Th B 100 M 10 M M Comparison between the two systems 53

100 Crore = 1 Billion 1 Million = 10 Lakh 1 Lakh = 100 thousand 1 Crore = 10 Million 10 Crore = 100 Million Guided 200 Million = ____ Crore 50 Million = ____ Crore 3 Crores = ____ million 60 lakhs = ____ million 7 lakhs = ____ thousand Place value Take the number 32154827. Place values of the digits are given below. Cr. 10 Lakhs Lakhs 10 Thou Thou Hundreds Tens Ones 7 x 1 = 7 2 x 10 = 20 8 x 100 = 800 4 x 1000 = 4,000 5 x 10000 = 50,000 1 x 1,00,000 = 1,00,000 2 x 10,00,000 = 20,00,000 3 x 1,00,00,000 = 3,00,00,000 54

3 2 1 5 4 8 2 7 = (3 × 1,00,00,000) + (2 × 10,00,000) + (1 × 1,00,000) + (5 × 10,000) + (4 × 1,000) + (8 × 100) + (2 × 10) + (7 × 1) Guided 1. Write in expanded form : 4 2 8 1 3 7 2 5 2. Write the place value of 5 in the numeral 35841276 Independent 1. Write in expanded form: a. 284317143 b. 18720581 2. Write the place value of the underlined digits in the following numerals: a. 3857346 b. 13748125 c. 8312743 Ordering numbers Ascending and descending order A. Write the following in ascending order. 2,81,378; 15, 30,747; 8,12,456; 9,21,478; 20,84,732 Step-1: a. Check the number of digits in each numeral. Number with the largest number of digits is the largest. b. If the number of digits is the same, the number with the largest digit in the highest place is the largest. Here, the largest number is 20,84,732, followed by 15,30,747, both having 7 digits. 55

In the remaining 6-digit numerals: 921478 is the largest, followed by 812456. 281378 is the smallest. Ascending order: 2 8 1 3 7 8 8 1 2 4 5 6 9 2 1 4 7 8 1 5 3 0 7 4 2 0 8 4 7 3 2 < < < < Guided 1. Arrange the following in ascending and descending order. a. 98247120, 84724420, 97413142, 69548312 b. 61821427, 7284991, 3472145, 92841742 Independent 1. Arrange the following in descending order: a. 4267128, 4266749, 3814521, 3841843 b. 12384715, 13402134, 1329487, 1498742 2. Arrange the following in ascending order: a. 763712, 434152, 875471, 2987152 b. 15804731, 1027431, 9814732, 10037152 56

Making numerals 1. Using the digits 9, 1, 7, 3, and 5, make the largest and smallest numerals. 2. Using digits 1, 7, and make the largest digit, digit and digit 3, 6 7 8 numerals. You can use each digit more than once if required. Largest 6 digit 7 7 7 7 3 1 7 digit 7 7 7 7 7 3 1 8 digit 7 7 7 7 7 7 3 1 Independent 1. Make the largest and smallest 7-digit numerals with the digits 9, 1, 5. 2. Make the largest and smallest 6-digit numerals with the digits 8, 7, 4, 3. Rounding numbers 1. Rounding off to nearest hundreds: a. 728 700 b. 672 700 Hint: If the ten’s digit is 5 or more than 5, add one to the hundreds place and replace the other places with 0. 2. Rounding off to the nearest thousands; 5 2 8 0 5 0 0 0 7 5 4 8 8 0 0 0 57

1. Round off the following to the nearest hundreds. a. 542 b. 1584 c. 2705 d. 603 2. Round off the following to the nearest thousands. a. 2753 b. 32504 c. 1805 d. 54025 Guided Independent 1. Round off to the nearest hundreds a. 763 b. 824 c. 908 d. 1268 e. 2507 f. 3284 2. Round off to the nearest thousands a. 1587 b. 2384 c. 15207 d. 12803 1. Write the following in words in Indian and International system. a. 54827064 b. 1528643 c. 72483143 d. 153381472 2. Re-write the above numerals, putting commas for Indian and international system. 3. Write in ascending order. 258731, 184347, 83438, 1538974 4. Put or > < 5318134 531843 328457 324875 153897 158397 58

5. Round off to the nearest thousand: 153743. 6. Make the largest 6-digit numeral using the digits 7, 9, 1, 3. You can use two digits two times each. 1. Circle the largest number: a. 753812, 785123, 87256, 92875 b. 15387, 18534, 9284, 10934 c. 11134, 11157, 113051, 11875 2. 40 M = ____ Cr 500 Cr = ____ M 700 Th = ____ L 50 Cr = ____ M 820 M = ____ Cr 59

Decode: if = 100 million, = 10 million, = 1 million, = 100 thousand, = 10 thousand, = 1 thousand, = 1 hundred, = 1 tens, = 1 ones. 1. What are the numbers given below? a. b. 2. Write the following in code language. a. Pay Rs.142,531,470 b. Issue a cheque for Rs.43,321,485 60

Problems to solve: 1. The population of 3 villages is 4,35,841, 5,82,140, and 9,25,340. Find the total population of the 3 villages, rounded off to the nearest lakhs without actual addition. 2. 10 = 10×1, 100 = 10×10 = 10 2 1000 = 10×10×10 = 10 , 3 10000 = 10×10×10×10 = 10 and so on. 4 Find out the number represented by the following notation. a. 3×10 + 4×10 + 5×10 + 2×10 + 6×10 + 7 7 6 4 3 b. 5×10 + 4×10 + 3×10 + 4×10 6 4 3 2 In the same way, 2×1=2, 2×2=2 , 2×2×2=2 , 2×2×2×2=2 etc. 2 3 4 3. Find the number given by: 2 + 2 + 2 + 2 + 2 + 1 5 4 3 2 61

Properties of Addition 1. 0 + any numeral = the numeral itself. Example: 0 + 150 = 150 2. Addition is commutative, which means that two numbers can be added in any order. Example: 23 + 54 = 54 + 23 3. Addition is associative, which means three or more numbers can be added in any order. Example: 2314 + 1372 + 184 = 2314 + 1840 +1372 = 1372 +1840 +2314 4. Addition is distributive. Example: 157 x ( 567 + 235) = (157 x 567) + (157 x 235) 62

Addition of 5, 6, 7, 8 digit numbers In a town, there are 45,71,754 males and 43,54,142 females. What is the total population of the town? Write in the place value chart according to commas 4 3 5 4 1 4 2 + 4 5 7 1 7 5 4 Cr. 10 L L 8 9 2 5 8 9 6 10Th Th H T U 1 Guided Independent A bank passbook shows three entries in 4 months, viz. Rs. 127456, Rs. 82151 and Rs, 7218465. What is the total amount shown by the entries? 1. A man has a house worth Rs 65,75,146, land worth Rs. 2,75,18,415 and a company worth Rs. 12,82,54,156. What is the total value of his property? 2. In an election, one candidate got 2,34,41 votes more than the other candidate who polled 43,25,100 votes. What is the total number of votes that were cast? 63

Digit Sum: Digit sum can be used to check whether addition of numbers is correct. Example: 4 5 7 6 5 + 3 8 9 8 6 digit sum = 4 + 5 +7 + 6 + 5 = 27 = 2 + 7 = 9 digit sum = 3 + 8 + 9 + 8 + 6 = 34 = 3 + 4 = 7 9 + 7 = 16 = 1 + 6 = 7 digit sum of the sum = 8 + 4 + 7 + 5 + 1 = 25 = 2 + 5 = 7 Digit sum of the sum = sum of the digit sums Practice: Add these and check whether addition is correct using digit sum 4 5 7 6 5 + 3 8 9 8 6 8 4 7 5 1 11 11 1. 4 5 8 7 1 2 5 2. 8 4 7 2 5 4 1 + 1 7 4 8 1 2 + 3 4 1 5 0 4 7 + 3 4 0 8 1 2 1 + 4 2 8 1 0 64

1. Add the following and check using digit sum: a. 278413 + 1784314 + 45371481 b. 148371 + 243842 + 3542154 2. Rice is packed in three trucks. The quantity of rice in the trucks is 1,28,157 kg, 2,84,75,472 kg and 2,53,457 kg respectively. Find the total quantity of rice in trucks. 3. In a company, Rs. 1,48,73,254 is paid as salary to the employees, Rs. 58,32,471 is spent for purchase of machinery and Rs, 24,32,154 is paid as income tax. What was the total amount spent? Find the missing digit a. 8 4 1 5 b. 2 2 7 4 + 3 6 8 6 4 5 8 2 2 5 1 3 4 3 8 1 4 6 8 8 2. Do the following additions and check with digit sum. a. 1 4 2 7 6 5 7 b. 1 8 2 3 7 8 + 4 1 5 8 4 3 4 + 4 8 6 8 4 7 + 1 8 2 4 3 8 + 1 5 3 9 4 7 Estimate: Estimate the sum of the following by rounding off to nearest lakhs 1. 1784315 + 2013471 + 813714 2. 2534716 + 1157146 + 2014716 65

Subtraction Properties of subtraction: 1. A number - 0 = the number itself 256 - 0 = 256 2. 0 – 256 not possible. 3. 256 – 256 = 0 4. 512 – 217 is possible. A smaller number can be subtracted from a larger number. But 217-512 is not possible Guided 1. Subtract 253714 from 575317 L 10 Th Th H T O 5 7 5 3 1 7 - 2 5 3 7 1 4 3 2 1 6 0 3 2. Find the difference between 312741 and 173835 L 10 Th Th H T O 3 1 2 7 4 1 - 1 7 3 8 3 5 1 3 8 9 0 6 13 4 11 3 17 1 0 2 10 11 66

1. Subtract 305874 from 612432 2. Find the difference between 154321 and 213081 3. How much more is Rs. 321574 than Rs. 253719 4. The population of a village is 2,49,30,165. 1,13,84,756 of it are females. What is the population of males in the village? Independent Subtraction from zeros Subtract 428431 from 1000000 10L L 10 Th Th H T O 1 0 0 0 0 0 0 - 4 2 8 4 3 1 5 7 1 5 6 9 Follow the rule - all from 9 and last from 10, reduce the first by 1. Last here means unit’s digit. 10 9 9 9 9 9 0 Independent 5 0 0 0 0 0 1 0 0 0 0 0 0 - 2 8 7 4 3 5 - 2 4 3 7 5 3 0 0 5 0 0 4 0 0 0 4 0 0 - 1 8 7 2 4 7 - 1 7 8 1 3 4 67

Subtraction: new method: e.g. 4 3 7 4 – 2 8 9 6 4 3 7 4 - 2 8 9 6 2 5 2 2 1. Find the difference between the digits, irrespective of their position – up or down. 2. If the bottom digits are larger, simply write the difference and put a bar over them. Do the following subtractions this way 2 5 2 2 (2-1) (9-5) (9-2) (10-2) 1 4 7 8 3. 4 7 4 2 1 5 4. 5 3 2 4 1 7 - 2 4 8 4 3 6 - 2 8 4 5 6 8 1. 3 2 5 4 3 2. 4 5 3 2 1 6 - 1 8 6 7 5 - 3 6 7 3 4 7 3. Use the rule: all from 9 and last from 10 to subtract the bar numbers from zeros and reduce the first by one. 4. Ans. : 1 4 7 8 Fun time Copy in your note book and do it. 68

1. Subtract 2 7 3 1 4 0 from 1 8 4 3 2 6 7. 2. Find the difference between 3 7 2 8 1 4 7 and 4 5 0 1 2 3 4. 3. Which is more – 6 7 2 0 8 1 or 1 3 1 2 1 4 7? By how much? 4. Out of Rs.37,54,75,143, Rs.9,73,84,968 was spent. How much money is remaining? 5. In a godown, there are 1,732,815 kgs of rice and 86,93,26 kgs of wheat. Which is more and by how much? 1. Find the missing digit a. 6 3 b. 4 2 3 3 4 2 3 4 7 5 5 6 7 3 6 5 2 5 2. Find the final answer in the chain. 4 5 8 7 1 6 – 2 7 3 9 2 4 + 1 7 8 9 7 2 – 5 8 4 9 3 + 1 5 2 8 9 3 – 6 9 7 8 3 = _____________________ 1. Conduct a team quiz. One team shows the placard with a number written on it. The other team reads the number as asked – in Indian system or international system. 2. One team gives addition & subtraction problems. The other team answers by showing the number cut outs. 3. Write your own word problems for addition and subtraction and give them to the other team for solving. 69

Multiplication Properties of Multiplication: 1. Any number multiplied by 0 gives zero. 125 × 0 = 0 2. Any number multiplied by 1 gives the number itself. 125×1 = 125 1 is the identity element for multiplication. 3. Multiplication is commutative. Example: 256×375 = 375×256 4. Multiplication is associative. Example: 256×375×180 = 180×256×375 = 375×180×256 5. Multiplication is distributive. Example: 210× (510+125) = (210×510) + (210×125) 70

Process of multiplication: Example 1: 4325 is called multiplicand 243 is the multiplier 243 = 200 + 40 + 3 First multiply 4325 by 3, then by 40, and then by 200. Add the results to get the product. 4325×3 = 12975 4325×40 = 173000 4325×200 = 865000 Adding Instead of putting zeros, the product in each step can be written starting from units, tens & hundreds place as shown Example 2: Multiplier – 2153 2153 = 2000 + 100 + 50 + 3 25385×3 = 76155 25385×50 = 1269250 Write it from 10’s place without 0 25385×100 = 2538500 Write it starting from 100’s place with out zero. 25385 × 2000 = 50770000 Write it starting from 1000’s place without the 3 zeros as shown. on the left Now, Add 4 3 2 5 x 2 4 3 1 2 9 7 5 1 7 3 0 0 * 10 5 0 9 7 5 + 8 6 5 0 * * 2 5 3 8 5 x 2 1 5 3 7 6 0 5 5 1 2 6 9 2 5 * 2 5 3 8 5 * * +5 0 7 7 0 * * * 5 4 6 5 3 9 0 5 71

1. Multiply the following: a. 5237 × 374 = b. 72615 × 432 = 2. 21754 copies of a book were printed. If each book has 329 pages in total, how many sheets of paper were used for printing all the copies? Guided Independent Do the following multiplications: 1. 3184 × 405 = 2. 12854 × 756 = 3. 43134 × 1234 = 4. 21374 × 3125 = 5. If one chair costs Rs.1254, what is the cost of 25813 such chairs? 6. A dairy sends out 23145 packets of milk daily. How many packets are produced (a) in a month (b) in a year? 7. There are 786 words in one page of a book. If there are 597 pages in the book, find the total number of words the book has. 72

Multiplication by splitting (using distributive property) 1. 425 × 98 = 425 × (100 – 2) = 425 × 100 – 425 × 2 = 42500 – 850 = 41650 2. 256 × 102 = 256 × (100 + 2) = 256 × 100 + 256 × 2 = 25600 + 512 = 26112 Independent Do the following using distributive property of multiplication: 1. 4186 × 99 2. 25754 × 101 3. 24307 × 102 4. 1567 × 998 Multiplication by regrouping (using associative property) 25 × 716 × 40 = 716 × (25 × 40) = 716 × 1000 = 716000 73

1. 50 × 756 × 80 = 2. 15 × 2154 × 20 = 3. 250 × 689 × 40 = 4. 20 × 1587 × 50 = Estimation: Estimate the product of the following by rounding off to the nearest highest place 1. 68375 × 1543 = 2. 72145 × 2740 = 3. 12385 × 3145 = 4. 253756 × 1483 = Activity 1: Multiplication by Napier rods Napier rods are multiplication rods. Units and tens places are separated by a slanting line. There are rods for multiplication tables up to 9. Problem: To multiply 3 and 5 In the multiplication table of 3, look for the 5 row. th It gives the product: 3 × 5 = 15 Guided 74 0 2 0 4 0 6 0 8 1 0 1 2 1 4 1 6 1 8 2 0 0 3 0 6 0 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 Table 2 Table 3 5th row

Problem: 23 × 5 Keep the multiplication table of 2 and 3 side by side and look for the 5 row in each. th The darkened row gives the required product. 1/0 1/5 1/(1+0)/5 Add the digits within the diagonals. You get 115. Thus, 23 multiplied by 5 = 115 0 2 0 4 0 6 0 8 1 0 0 3 0 6 0 9 1 2 1 5 Problem: 32 × 6 Ans. 192 1 2 3 4 5 0 2 0 4 0 6 0 8 1 0 0 3 0 6 0 9 0 2 1 5 6 1 2 1 8 Problem: 32 × 23 Ans: 06 36 1 Carry over 1 to 6 Ans. 736 2 8 1 1 1 2 3 0 2 0 4 0 6 0 3 0 6 0 9 13 0 4 0 6 0 6 0 9 6 0 9 2 2 6 3 6th row 5th row 2 3 5 6 2 3 2nd row 3rd row Extended activity: Find the products of the following using Napier rods. 1. 25 × 5 = 2. 43 × 4 = 3. 23 × 25 = 4. 35 × 24 = 75

1. 112 × 238 = 238 × (12 + ) 2. 135 × = 357 × 3. 127 × × 328 = 287 × 328 × = 127 × × 287 4. 200 × 30 × 5 = _______ 5. 120 × 30 × 50 = _______ 6. 23 × 32 = 32 × 23 (true/false) 7. 18 × 40 = 18 × 4 × 10 (true/false) 8. 127 × 40 × 50 = 127 × 2 × 100 (true/false) 9. Do the following multiplications. a. 2837 × 1485 = b. 43847 × 285 = c. 38145 × 2405 = d. 47281 × 389 = 10. Multiply by regrouping (use associative and distributive properties) a. 123 × 105 = b. 25 × 80 × 185 = c. 344 × 998 = d. 152 × 110 = 76

Frame word problems on multiplication using the numbers given below. 1) 11384; 249 2) 381045; 254 3) 29, 81, 341; 485 2. 998 × 999 Write the numbers and differences from 1000. Multiply the differences. Cross subtract differences from numbers Ans : 997002 Now do these multiplications 1. 96 × 98 2. 99 × 99 3. 998 × 998 4. 9999 × 9989 998 - 002 999 - 001 997 / 002 98 - 02 97 - 03 95 / 06 Multiplication without multiplying: 1. 98 × 97 98 and 97 are nearer to 100. 98 – 02 Difference of 98 from 100 is 2. 97 – 03 Difference of 97 from 100 is 3. 06 Write the numbers as shown on the right. 02 and 03 are differences. Multiply the differences and cross subtract the number. Difference on cross subtracting either of the numbers is 95. (98 – 03 or 97 – 02). Thus, answer is 9506. 77

Division Test of divisibility: First, let’s revise what we learnt in the previous class. 1. All even numbers, or numbers with 0, 2, 4, 6, 8 in the unit's place are divisible by 2. 2. If the sum of the digits of a number is divisible by 3, the number is divisible by 3. 3. If the digits in the unit's and ten's place taken together are divisible by 4, the number is divisible by 4. 4. If the unit’s place is 5 or 0, it is divisible by 5. 5. If a number is divisible by both 3 and 2, it is divisible by 6. 6. If the sum of digits of a number is divisible by 9, the number is divisible by 9. Now, we shall learn the test of divisibility for 7, 8 and 11. Divisibility by 7 If twice the digit in the unit's place is subtracted from the rest of the number, and the resulting number is divisible by 7, the number is divisible by 7. 78

Example: 525 Digit in the unit’s place = 5 Twice of 5 = 10 Subtract 10 from 52 52 – 10 = 42 42 is divisible by 7. Therefore, 525 is divisible by 7. See whether the following are divisible by 7. 112, 455, 434, 623, 1286 Divisibility by 8 If the number formed by the last 3 digits of a number taken together is divisible by 8, the number is divisible by 8. Example: 5128 The last 3 digits of the number form 128. 128 is divisible by 8. Thus, 5128 is divisible by 8. Check whether 7256 and 8512 are divisible by 8. Divisibility by 11 If the sum of alternate digits of a number is the same, or the difference of the sum is a multiple of 11, the number is divisible by 11. Example: 14641 sum of alternate digits: 1 + 6 + 1 = 8; 4 + 4 = 8 There are same Therefore, 14641 is divisible by 11. Similarly, 2378562: 2 + 7 + 5 + 2 = 16: 3 + 8 + 6 = 17 The sums are different. Their difference is 1, which is not divisible by 11. Therefore, the number is not divisible by 11 79

Practice: Test the divisibility of the numbers by 8, 7 and 11 Number Divisibility by 7 Divisibility by 8 Divisibility by 11 8344 26972 70169208 5533 32716 385 693 Properties of division: 1. 256 1= 256 2. 256 256 = 1 3. 0 256 = 0 4. 256 0 = Not possible 5. Dividend = Divisor x Quotient + Remainder Division by 10, 100, 1000 etc When divided by 10: 3287 = Quotient 328, Remainder 7 Written as decimal 328.7 80

When divided by 100 : 3287 = Quotient 32, Remainder 87 In decimal form Q = 32.87 When divided by 1000 3287 = Quotient 3, Remainder 287 In decimal form Q = 3.287 Independent 1. 32845 1 = ÷ __________________ 2. 0 158 = ÷ ____________________ 3. Divisor = 12, Quotient = 8, Remainder 5. Dividend = ____________ 4. When 3581 is divided by a number, the quotient is 596 and the remainder is 5. The divisor is _________ 5. When 1125 is divided by a number, the quotient is 160 and the remainder is 5. The divisor is ___________________ Division by 3 and 4 digit numbers Example: 17285 1 ÷ 21 1. Observe the number formed by the first 3 digits of 17285. It is 172. 172 is bigger than 121. So, we can divide it. Carry on the division as shown. 81

2. Bring down 8. Multiply 121 by 2, 3, 4…. till you get a number equal to 518 or just less than that. Incase it is 4. then subtract 518 - 484 3. Bring down 5. Do the division of 345 Thus, Quotient is 142; Remainder is 103. Checking: 121 x 142 + 103 = 17285 1 7 2 8 5 1 121 1 2 1 5 1 4 2 1 7 0 2 1 3 9 3 3 1072 - 3 2 1 6 1 0 0 1 0 - 9 6 4 8 3 6 2 2 - 3 2 1 6 4 0 6 1 - 3 2 1 6 0 8 4 5 Example 2: 4217021 1072 ÷ 1. Take the first 4 digits. The number formed is bigger than 1072. Thus, we can divide it. 2. Multiply 1072 by 2, 3, 4 ...... till you get a number equal to or just less than 4217. 1072 x 3 = 3216 1072 x 4 = 4288 4288 is bigger than 4217 Take 1072 x 3 3. Bring down 0. 4. Try multiplying 1072 by 9, 8, 7…. till a suitable number is obtained 5. Carry on the division. In the end, you will get Q = 3933, R = 845. Checking: 1072 x 3933 + 845 = 4217021 1 7 2 8 5 1 4 2 121 1 2 1 5 1 8 - 4 8 4 3 4 5 - 2 4 2 10 3 82

Solve a. 70329 452 ÷ = b. 7218367 2387 ÷ = Guided Independent Solve the following and check the answer: a. 18392 48 ÷ = b. 73602 123 ÷ = c. 4567890 2527 ÷ = 1. Frame word problems for the following division facts. a. 275 25 = b. 10440 29 = c. 3472 14 = 2. If dividend is 2496, quotient is 118, and remainder is 18, find the divisor 3. Dividend: 4328; Quotient: 206; remainder: 2. Find divisor. 83

1. Find whether the following are divisible by 11: a. 638151 b. 95469 c. 2839155 d. 6570 e. 82335 2. Which of the following are divisible by 8? a. 32456 b. 32468 c. 92417 d. 13872456 3. Divide and check the answers: a. 828175 428 b. 5728612 1345 c. 931258 1754 d. 13680 765 4. The cost of 215 sofas is Rs. 325944. What is the cost of 1 sofa? 5. A stadium which can seat 1656561 people has 487 rows. How many seats are there in each row? 6. Product of two numbers is 6225. One of the numbers is 83. Find the other. 84

Highest Common Factor (HCF) It is also called the Greatest Common Divisor (GCD). HCF is the highest among the common factors of two or more numbers. Methods of finding HCF 1. Common factor method. Find the common factors and choose the largest among them. To find HCF of 24 and 30: Factors of 30 30 = 3 × 10, 2 × 15, 1 × 30, 6 × 5 Factors of 24 24 = 1 × 24, 2 × 12, 3 × 8, 4 × 6 Thus, factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24 And, factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30 Common factors of 24 and 30 = 1, 2, 3, 6 Therefore, highest common factor (HCF) is = 6 Find the HCF of: a. 12, 24 b. 12, 15, 20 Guided Independent 1. Find the HCF of the following: a. 24, 25 b. 32, 18 c. 63, 42 d. 12, 36, 30 e. 15, 20, 45 85

2. Prime factorization method Example: To find HCF of 24 and 18 24 = 2 × 2 × 2 × 3 18 = 2 × 3 × 3 24 6 2 3 4 2 2 Thus, common prime factors = 2 × 3 = 6 Prime factorization can also be done by division. Product of the divisors = 2 × 3 = 6. Thus, HCF = 6 3. Division Method Divide the numbers by a common divisor. Carry on the division till both the quotients have a common factor. Example: 16, 32 HCF = 2 × 8 = 16 18 2 9 3 3 16, 32 2 8, 16 8 1, 2 Guided Find HCF of the following by prime factorization. a. 45, 36 b. 48, 72, 36 86

1. Find HCF of the following by prime factorization a. 18, 42 b. 48, 72, 120 c. 144, 216 d. 90, 120, 150 Independent HCF by long division – Division algorithm First divide the bigger number by the smaller number. Then divide the divisor by the remainder. Go on dividing till the remainder is zero. The last divisor is the HCF Example 1: 48, 60 HCF = 12 Example 2: To find HCF of 624, 264 HCF = 24 48) 60 ( 1 48 12) 48 (4 48 0 Guided Find HCF of the following by division algorithm: a. 144, 264 b. 25, 90 c. 24, 45, 57 (Find HCF of 57 and 45; divide 24 by the HCF till the remainder is zero.) 87 264) 624 (2 - 528 96) 264 (6 - 192 72) 96 (1 - 72 24) 72 (3 - 72 0

Find HCF of the following by division algorithm. a. 72, 192 b. 48, 108, 112 c. 84, 51 d. 18, 45, 54 e. 264, 840, 384 HCF of co-prime numbers Find HCF of 15 and 28. Independent Activity : To demonstrate division algorithm : Why do we divide the bigger number by the smaller number to find the HCF by long division method? It will be made clear by the following activity. To find the longest piece that can be cut from two strips of different lengths. 1. Take two strips of cloth, one of length 21 cm and the other of length 30 cm. 2. Keep the 21 cm strip over the 30 cm strip. Cut off the extra length. HCF = 1 Practice: Show that the following pairs are co-prime. a. 23, 25 b. 18, 41 c. 71, 75 15) 28 (1 15 13) 15 (1 13 2) 13 (6 12 1) 2 (2 2 0 88

3. Take the cut off piece, which is 9 cm, and keep it over the 21 cm strip. You can measure 9 cm two times from the 21 cm strip. 3 cm is left out. Cut off the 3 cm piece. 4. Keep the 3 cm piece on the 9 cm piece. It can measure the 9 cm piece completely. 5. The longest piece that can be cut from both the strips is 3 cm. Thus, HCF of 21 and 30 is 3. Extended activity: 30 cm 21 cm 9 cm 9 cm 3 cm 21 cm 9 cm 3 cm Find the longest strip that can be made from both 30 and 45 cm long strips. 1. Find the HCF of the following by prime factorization. a. 96, 128 b. 84, 72 2. Find the HCF of the following by common factor method. a. 20, 45 b. 45, 30, 27 c. 42, 70 d. 25, 90, 75 3. Find the HCF of the following by division algorithm. a. 72, 192 b. 48, 128 c. 360, 540, 810 4. Find the HCF of the following by all the three methods. a. 42, 48, 90 b. 144, 192 3 cm 3 cm3 cm 9 cm 89

1. Find the largest number that divides 735 and 1155 without leaving any remainder. 2. Find the longest piece that can be cut from cloths of length 80m, 96m and 48m. 3. What is the side of square tiles required for a room with dimension 9ft by 12ft. to fit in without cutting the tiles? 4. Find the largest number that can divide 280, 413 and 666 leaving remainder 4, 5, 6 respectively. (Subtract the remainders from the numbers and then find HCF) Least Common Multiple (LCM) Consider the number 12. It comes in the multiplication tables of 2, 3, 4, and 6. Thus, we say that 12 is a multiple of 2, 3, 4 and 6 We can also say 12 is a common multiple of 2, 3, 4 and 6. Let’s see the common multiples of only 2 and 3 from the grid. Common multiples of 2 and 3 from the grid are 6, 12 and 18. We can see that 6 is the lowest common multiple of 2 and 3. Thus, we say that LCM of 2 and 3 is 6. 90

To find LCM of 2, 3 and 4: Multiples of 2 = 2, 4, 6, 8, 12, 14, 16, 18, 20, 22, 24, ….. Multiples of 3 = 3, 6, 9, 12, 15 18, 21, 24, … Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, ….. Common multiples of 2, 3 & 4 = 12, 24, …. Least common multiple = 12. Thus, LCM of 2, 3, 4 = 12 To find LCM of 6, 8, 12: Multiples of 6 = 6, 12, 18, 24 , 30, 36, 42, 48 , … Multiples of 8 = 8, 16, 24 , 32, 40, 48 , ….. Multiples of 12 = 12, 24 , 36, 48, ….. Common multiples = 24, 48,….. Least common multiple = 24. Thus, LCM of 6, 8, 12 = 24 Independent Find the LCM of the following by the method of common multiples a. 16, 8, 6 b. 10, 5, 20 c. 4, 6, 12 91

LCM by division method Problem: To find LCM of 16, 12, 24: 1) Find a number that can divide all the three. 4 can divide all the three. 2) Find a number which can divide all the three quotients. If not found, look for a number that can divide at least 2 of them. Here, you can choose either 2 or 3. Divide by 2. 3) Divide the quotients by 3. LCM = Product of the divisors and quotients. LCM = 4 × 2 × 3 × 2 × 1 × 1 = 48 16, 12, 24 4 4, 3, 6 2 2, 3, 3 3 2, 1, 1 Guided 1. Find the LCM of the following by division method a. 27, 45 b. 48, 72 c. 22, 24, 36 d. 30, 45, 60 e. 15, 24, 36 LCM of co-prime numbers Example: Find the LCM of 11 and 15. There are no common factors for these two numbers. Therefore, their LCM is 11 × 15 = 165 LCM of co-prime numbers is the product of the numbers. 92

Find the LCM of: a. 4, 5 b. 15, 32 c. 8, 5 d. 2, 15 e. 3, 7, 2 Independent Product of LCM & HCF Example: Let’s take numbers 10 and 15 HCF of 10 and 15 = 5 LCM of 10 and 15 = 30 Product of LCM and HCF = LCM × HCF = 5 × 30 = 150 Product of the numbers = 10 × 15 = 150 Product of LCM and HCF = Product of the numbers. Guided 1) Product of LCM & HCF of two numbers is 240. One of the numbers is 30. Find the other number. 2) Product of two numbers is 120. Their HCF is 3. Find the LCM. 93

1) Product of LCM and HCF of two numbers is 300. One of the numbers is 15. Find the other number. Also find their HCF and LCM. 2) Product of two numbers is 120. Their HCF is 2. Find the LCM. 3) Product of 2 numbers is 1350. Their LCM is 90. Find the HCF. Independent 1. Find 5 common multiples of 5 and 6. 2. Find 5 common multiples of 2, 3, and 5. 3. Find the LCM of the following by division method. a. 18, 15, 20 d. 16, 32, 64 b. 14, 20, 28 e. 15, 45 c. 21, 42, 84 f. 15, 12, 90 4. Find the smallest 3-digit number which is a multiple of 2 and 3. 1. Find the smallest number divisible by 15, 12, and 20. 2. Find the smallest number divisible by 2, 3, 4 and 5. 3. Find the least number divisible by 15, 20, and 24, leaving remainder 5 each time. (Find LCM and add 5) 4. 3 bells in a temple ring at intervals of 12, 15, and 18 minutes. When will the three ring together for the first time? 94

Symmetry and Patterns Symmetry Shapes that can be divided into mirror images or shapes that can be regenerated by rotating through certain angles are said to be symmetrical. Symmetrical objects have: 1) An axis of symmetry, if the object is a plane shape. 2) A plane of symmetry, if the object is a solid 3) An angle of symmetry or rotational symmetry Axis of Symmetry when sphere is cut vertically Plane when a cylinder is cut vertically Axis of symmetry Plane of symmetry Axis of symmetry When a cone is cut vertically Plane of symmetry Plane Plane 95

Experiment: Take cutouts of a rectangle, a square, an equilateral triangle, a pentagon and a hexagon. By folding along the dotted line, find the axes of symmetry for the figures A rectangle has a vertical and a horizontal axis of symmetry A square has 1 vertical, 1 horizontal and 2 diagonal axes of symmetry No axes of symmetry for a scalene triangle Isosceles triangle has one vertical axis of symmetry An equilateral triangle has 3 axes of symmetry. 96

There are 5 axes of symmetry for a pentagon. Four axes of symmetry are drawn for a hexagon. Can you find more? There are 4 planes of symmetry for a cube. There are 2 planes of symmetry for a cuboid. There are 2 planes of symmetry for a cylinder drawn here. There can be more vertical planes of symmetry. There are infinite number of planes of symmetry for a sphere and infinite number of axes of symmetry for a circle. 97

1. Draw the axes of symmetry for the following figures, if they have any. 2. Draw the line of symmetry for the following figures, if they have any. 3. Complete the other half 98

Rotational symmetry Total angle 360 4 4 You had to rotate the figure 4 times to get back to the original position. However, each of the rotations brought the shape back to its position to fit into the outline. The number of times a figure has to be rotated to come back to its original orientation is called the order of symmetry. From the above activity, we can conclude that the order of rotation for a square is 4. Angle of rotation = = = 90 0 Activity 1: Take cutouts of a square, a rectangle, and an equilateral triangle, cutout from a cardboard. a. Keep the square on the paper. Mark the vertices as ABCD. Draw its outline, Rotate the cutout, till it falls exactly on the outline Note the position of the letters ABCD. Have they rotated? Rotate again, till it coincides with the outline. Note the position of the letters again. Rotate it again till it coincides with the outline. Note the letters. Have they come back to their original position? A B D C A B C D A B D C A B D C A B D C 99

It means that if the square is rotated through 90 , it will come back to 0 its original position. However, its vertices will be changed. 360 4 The order of rotation = 2. Angle of rotation = = 180 0 For a rectangle, the angle of rotation is 180 . It means that only when 0 it is turned through 180 , it will fit into its original position. 0 A B D C A B C D A B D C A B D C A B D C Equilateral triangle: The triangle has to be rotated 3 times to get back to its original orientation. Thus, order of rotation = 3. A B C A B C A B C A C B Angle of rotation = = 120 0 For every 120 rotation, the triangle will fit into the outline drawn. 0 360 4 100 Activity : 1 Try the same with the rectangle cut out 2 times the cut was coinciding with the original positions