CHAPTER 1 Knowing Our Numbers POINTS FOR QUICK REVISION l Natural Numbers: To count and calculate we use numbers 1, 2, 3, 4,... which are called counting numbers or natural numbers. l There is no greatest natural number i.e. these are uncountable. l Number 0 together with natural numbers gives us 0, 1, 2, 3,... are called whole numbers and denoted by W = {0, 1, 2,...} l Ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 which are used to represent any number are called digits or figures. l While forming numbers from given digits we should take care of condition(s) given. l Place value of a digit in a number = Its face value position value. l 1 km = 1000 m 1 m = 100 cm 1 cm = 1000 mm 1 kg = 1000 gm 1 litre = 1000 ml l 1 million = 10 lakh 10 million = 1 crore 100 million = 100 crore or 1 Arab. l Estimation means approximating a quantity to an accuracy required. l In calculation use of brackets allow us to carry out more than one number operation. l Roman numerals I V X L C D M Hindu-Arabic numerals 1 5 10 50 100 500 1000 l There is no zero in Roman system. Roman system does not use place value. l If a bar is placed over a numeral it is multiplied by 1000 i.e. V = 5000 and L = 50000. l l Palindromic Number is a number that is same whether read backwards or forward for example: 575. In India we use two major numeration systems Indian system and International system. Use of commas helps in reading and writing large numbers.
2 MATHEMATICS VI l In the Indian system of numeration we have commas after 3 digits starting from the right and thereafter after every 2 digits. The commas after 3, 5 and 7 digits separate thousand, lakh and crore respectively. l In the International system of numeration commas are placed after every 3 digits starting from the right. The commas after 3 and 6 digits separate thousand and million respectively. NCERT TEXT BOOK QUESTIONS (Page 2) Can you instantly find the greatest and the smallest numbers in each row? Q. 1. 382, 4972, 18, 59785, 750 Sol. 59785 is the greatest and 18 is the smallest. Q. 2. 1473, 89423, 100, 5000, 310 Sol. 89423 is the greatest and 100 is the smallest. Q. 3. 1834, 75284, 111, 2333, 450 Sol. 75284 is the greatest and 111 is the smallest. Q. 4. 2853, 7691, 9999, 12002, 124 Sol. 12002 is the greatest and 124 is the smallest. Q. 5. Was it easy? Why was it easy? Sol. Yes, it was easy. We just looked at the number of digits and obtained the answer. The number in which the number of digits was greatest, was the greatest number. If in two numbers the number of digits was the same, then the number in which on moving to the left the leftmost digit was greater, was the greater number. If the leftmost digits were also the same, than we saw the next digit. The number in which the next digit was greater, was the greater number. If requirement be, we proceeded next to next digits. (Page 2) Q. 1. Find the greatest and the smallest numbers. (a) 4536, 4892, 4370, 4452 (b) 15623, 15073, 15189, 15800 (c) 25286, 25245, 25270, 25210 (d) 6895, 23787, 24569, 24659 Make five more problems of this kind and give to your friends to solve. Sol. (a) The greatest number is 4892. The smallest number is 4370. (b) The greatest number is 15800. 1
KNOWING OUR NUMBERS 3 The smallest number is 15073. (c) The greatest number is 25286. The smallest number is 25210. (d) The greatest number is 24659. The smallest number is 6895. Five more problems of this kind are given below. 1. Find the greatest and the smallest numbers. 5936, 5079, 5788, 5672 Greatest : 5936 Smallest : 5079 2. Find the greatest and the smallest numbers. 31972, 31729, 31649, 31545 Greatest : 31972 Smallest : 31545 3. Find the greatest and the smallest numbers. 42375, 42367, 42329, 42338 Greatest : 42375 Smallest : 42329 4. Find the greatest and the smallest numbers. 31896, 31897, 31899, 31890 Greatest : 31899 Smallest : 31890 5. Find the greatest and the smallest numbers. 25279, 20791, 3987, 3978 Greatest : 25279 Smallest : 3978 CHAPTER 1 Q. 1. Use the given digits without repetition and make the greatest and smallest four digit numbers? (a) 2, 8, 7, 4 (b) 9, 7, 4, 1 (c) 4, 7, 5, 0 (d) 1, 7, 6, 2 (e) 5, 4, 0, 3 (Hint. 0754 is a 3-digit number.) Sol. (a) Greatest four digit number = 8742 ; Smallest four digit number = 2478 (b) Greatest four digit number = 9741 ; Smallest four digit number = 1479 (c) Greatest four digit number = 7540 ; Smallest four digit number = 4057 (d) Greatest four digit number = 7621 ; Smallest four digit number = 1267 (e) Greatest four digit number = 5430 ; Smallest four digit number = 3045 Q. 2. Now make the greatest and the smallest four digit numbers by using any one digit twice. (a) 3, 8, 7 (b) 9, 0, 5 (c) 0, 4, 9 (d) 8, 5, 1 (Hint. Think in each case which digit will you use twice.) (Pages 3 4) Sol. (a) Greatest four digit number = 8873 ; Smallest four digit number = 3378 (b) Greatest four digit number = 9950 ; Smallest four digit number = 5009 (c) Greatest four digit number = 9940 ; Smallest four digit number = 4009 (d) Greatest four digit number = 8851 ; Smallest four digit number = 1158 In the greatest number, the greatest digit is used twice while in the smaller number, the smallest digit is used twice. Q. 3. Make the greatest and the smallest 4- digit numbers using any four different digits, with conditions as given. (a) Digit 7 is always at ones place Greatest 9 8 6 7 Smallest 1 0 2 7 (Note, the number cannot begin with the digit 0. Why?) (b) Digit 4 is always at tens place Greatest 4 Smallest 4
4 MATHEMATICS VI (c) Digit 9 is always at hundreds place Greatest 9 Smallest 9 (d) Digit 1 is always at thousands place Greatest 1 Smallest 1 Sol. (b) Greatest 9 8 4 7 Smallest 1 0 4 2 (c) Greatest 8 9 7 6 Smallest 1 9 0 2 (d) Greatest 1 9 8 7 Smallest 1 0 2 3 The number cannot begin with the digit 0 as then number will actually be a 3-digit number. Q. 4. Take two digits, say 2 and 3. Make 4-digit numbers, using both the digits equal number of times. Which is the greatest number? Which is the smallest number? How many different numbers can you make in all? Sol. Four digit numbers formed by using two digits 2 and 3 equal number of times are : 2233, 2323, 2332, 3223, 3232, 3322 The greatest number is 3322 The smallest number is 2233 We can make 6 different numbers in all. (Page 4) Q. Think of five more situations where you compare three or more quantities. Sol. Five more situations of this type are as follows: 1. The weights of 3 students Manish, Krishna and Yash of section A of class VI are respectively 25 kg, 20 kg and 18 kg. 2. The number of runs scored in an inning by four players Ram, Rahim, Mohan and Shubham is respectively 30, 26, 46 and 15. 3. The heights of five girls are respectively 130 cm, 150 cm, 145 cm, 167 cm and 158 cm. 4. The monthly incomes of four persons Mukut, Girraj, Dandayal and Vishnu are respectively ` 10000, ` 8000, ` 9000 and ` 7500. 5. The total number of students in three schools A, B and C are 1900, 2000 and 1640 respectively. (Page 5) Q. 1. Arrange the following numbers in ascending order. (a) 847, 9754, 8320, 571 are : (b) 9801, 25751, 36501, 38802 Sol. (a) The given numbers in ascending order 571, 847, 8320, 9754 (b) The given numbers in ascending order are : 9801, 25751, 36501, 38802 Q. 2. Arrange the following numbers in descending order. (a) 5000, 7500, 85400, 7861 (b) 1971, 45321, 88715, 92547
KNOWING OUR NUMBERS 5 Sol. (a) The given numbers in descending order are : 85400, 7861, 7500, 5000 (b) The given numbers in descending order are : 92547, 88715, 45321, 1971 Q. 3. Make ten such examples of ascending/ descending order and solve them. Sol. Ten such examples of ascending/ descending order are as follows: Arrange the following numbers in ascending order: Ex. 1. 958, 9865, 9431, 682 Ex. 2. 736, 8643, 7210, 460 Ex. 3. 625, 7532, 6101, 350 Ex. 4. 514, 6421, 5012, 240 Ex. 5. 403, 5310, 4001, 130 Sol. The numbers arranged in ascending order are as follows: 1. 682, 958, 9431, 9865 2. 460, 736, 7210, 8643 3. 350, 625, 6101, 7532 4. 240, 514, 5012, 6421 5. 130, 403, 4001, 5310 Arrange the following numbers in descending order: Ex. 1. 6000, 8600, 86500, 8972 Ex. 2. 7000, 6400, 74300, 6750 Ex. 3. 2892, 56432, 99826, 93658 Ex. 4. 2760, 34210, 66604, 81436 Ex. 5. 1650, 23101, 55513, 70328 Sol. The numbers arranged in descending order are as follows: 1. 86500, 8972, 8600, 6000 2. 74300, 7000, 6750, 6400 3. 99826, 93658, 56432, 2892 4. 81436, 66604, 34210, 2760 5. 70328, 55513, 23101, 1650 CHAPTER 1 (Page 6) Q. 1. Read and expand the numbers wherever there are blanks. Number Number Name Expansion 20000 Twenty thousand 2 10000 26000 Twenty six thousand 2 10000 + 6 1000 38400 Thirty eight thousand four hundred 3 10000 + 8 1000 + 4 100 65740 Sixty five thousand seven hundred 6 10000 + 5 1000 + 7 100 + 4 10 forty 89324 Eighty nine thousand, three hundred 8 10000 + 9 1000 + 3 100 + 2 10 + 4 1 twenty four 50000 41000 47300 57630 29485 29085 20085 20005
6 MATHEMATICS VI Sol. 50000 Fifty thousand 5 10000 41000 Forty one thousand 4 10000 + 1 1000 47300 Forty seven thousand three hundred 4 10000 + 7 1000 + 3 100 57630 Fifty seven thousand six hundred thirty 5 10000 + 7 1000 + 6 100 + 3 10 29485 Twenty nine thousand four hundred 2 10000 + 9 1000 + 4 100 + 8 10 + 5 1 eighty five 29085 Twenty nine thousand eighty five 2 10000 + 9 1000 + 8 10 + 5 1 20085 Twenty thousand eighty five 2 10000 + 8 10 + 5 1 20005 Twenty thousand five 2 10000 + 5 1 Q. 2. Write five more 5-digit numbers, read them and expand them. Sol. Number Number Name Expansion 30000 Thirty thousand 3 10,000 52000 Fifty two thousand 5 10,000 + 2 1,000 36400 Thirty six thousand 3 10,000 + 6 1,000 four hundred + 4 100 68940 Sixty eight thousand 6 10,000 + 8 1,000 nine hundred forty + 9 100 + 4 10 24579 Twenty four thousand 2 10,000 + 4 1,000 five hundred seventy nine + 5 100 + 7 10 + 9 1 (Page 7) Q. Read and expand the numbers wherever there are blanks. Number Number Name Expansion 3,00,000 Three lakh 3 1,00,000 3,50,000 Three lakh fifty thousand 3 1, 00,000 + 5 10,000 3,53, 500 Three lakh fifty three thousand 3 1,00,000 + 5 10,000 + 3 1000 + 5 100 five hundred 4,57,928 4,07,928 4,00,829 4,00,029 Sol. 4,57,928 Four lakh fifty seven thousand 4 1,00,000 + 5 10,000 + 7 1000 nine hundred twenty eight + 9 100 + 2 10 + 8 1 4,07,928 Four lakh seven thousand nine 4 1,00,000 + 7 1000 + 9 100 hundred twenty eight + 2 10 + 8 1 4,00,829 Four lakh eight hundred twenty nine 4 1,00,000 + 8 100 + 2 10 + 9 4,00,029 Four lakh twenty nine 4 1,00,000 + 2 10 + 9 1
KNOWING OUR NUMBERS 7 Q. 1. What is 10 1 =? 2. What is 100 1 =? 3. What is 10,000 1 =? 4. What is 1,00,000 1 =? 5. What is 1,00,00,000 1 =? (Hint: Use the said pattern.) Q. 1. Give five examples where the number of things counted would be more than 6-digit number. Sol. 1. Number of people in U.P. state. 2. Number of copies of Ram Charit Manas published each year. 3. Number of grains in a sack full of wheat. 4. Number of tickets sold in 2002 2003 in U.P. 5. Number of saving account holders in nationalised banks in U.P. Q. 2. Starting from the greatest 6-digit number, write the previous five numbers in descending order. Sol. The largest 6-digit number is 9,99,999. The previous five numbers in descending order are (Page 8) Sol. 1. 10 1 = 9 2. 100 1 = 99 3. 10,000 1 = 9,999 4. 1,00,000 1 = 99,999 5. 1,00,00,000 1 = 99,99,999 (Page 8) (Page 11) 9, 99, 998 ; 9, 99, 997 ; 9, 99, 996 ; 9, 99, 995 ; 9, 99, 994. Q. 3. Starting from the smallest 8-digit number write the next five numbers in ascending order and read them. Sol. The smallest 8-digit number is 1,00,00,000. The next 5 numbers in ascending order are : 1,00,00,001 One crore one 1,00,00,002 One crore two 1,00,00,003 One crore three 1,00,00,004 One crore four 1,00,00,005 One crore five Q. 1. Read these numbers. Write them using placement boxes and then write their expanded forms. (i) 475320 (ii) 9847215 (iii) 97645310 (iv) 30458094 (a) Which is the smallest number? (b) Which is the greatest number? (c) Arrange these numbers in ascending and descending order. Sol. S.No. Number T Cr Cr T La La T Th Th H T O (i) 475320 4 7 5 3 2 0 CHAPTER 1 Number name Four lakh, seventy five thousand, three hundred twenty Expansion 4 1,00,000 + 7 10,000 + 5 1000 + 3 100 + 2 10 + 0 (Contd.)
8 MATHEMATICS VI (ii) 9847215 9 8 4 7 2 1 5 Ninety eight lakh, forty seven thousand, two hundred fifteen 9 10,00,000 + 8 1,00,000 + 4 10,000 + 7 1000 + 2 100 + 1 10 + 5 (iii) 97645310 9 7 6 4 5 3 1 0 Nine crore, seventy six lakh, forty-five thousand, three hundred ten 9 1,00,00,000 + 7 10,00,000 + 6 1,00,000 + 4 10,000 + 5 1,000 + 3 100 + 1 10 + 0 (iv) 30458094 3 0 4 5 8 0 9 4 Three crore, four lakh, fifty eight thousand ninety four 3 1,00,00,000 + 4 1,00,000 + 5 10,000 + 8 1,000 + 9 10 + 4 (a) The smallest number is 475320. (b) The greatest number is 97645310. (c) The numbers arranged in ascending order are as follows : 475320, 9847215, 30458094, 97645310 The numbers arranged in descending order are as follows : 97645310, 30458094, 9847215, 475320. Q. 2. Read these numbers (i) 527864 (ii) 95432 (iii) 18950049 (iv) 70002509. (a) Write these numbers using placement boxes and then using commas in Indian as well as International System of Numeration. (b) Arrange these in ascending and descending order. Sol. (a) (I) Using Placement Boxes S.No. Number T Cr Cr T La La T Th Th H T O Number name (i) 527864 5 2 7 8 6 4 Five lakh twenty seven thousand eight hundred sixty four (ii) 95432 9 5 4 3 2 Ninety five thousand four hundred thirty two (iii) 18950049 1 8 9 5 0 0 4 9 One crore eighty nine lakh fifty thousand forty nine (iv) 70002509 7 0 0 0 2 5 0 9 Seven crore two thousand five hundred nine
KNOWING OUR NUMBERS 9 (II) Using Commas Number Number using commas in Number using commas Indian system of numeration in International system of numeration (i) 527864 5,27,864 527,864 (ii) 95432 95,432 95,432 (iii) 18950049 1,89,50,049 18,950,049 (iv) 70002509 7,00,02,509 70,002,509 (b) Arranged in ascending order, these numbers are as follows : 95432, 527864, 18950049, 70002509 Arranged in descending order, these numbers are as follows : 70002509, 18950049, 527864, 95432. Q. 3. Take three more groups of large numbers and do the exercise given above. Sol. Three more groups of large numbers are as follows: I. 637534, 86543, 27860038 II. 748645, 97654, 38970049 III. 526423, 75432, 16750027 (I) (a) (1) Using Placement Boxes CHAPTER 1 S. No. Number T Cr Cr T La La T Th Th H T O (i) 637534 6 3 7 5 3 4 Number name Six lakh thirty seven thousand five hundred thirty four (ii) 86543 8 6 5 4 3 Eighty six thousand five hundred forty three (iii) 27860038 2 7 8 6 0 0 3 8 Two crore seventy eight lakh sixty thousand thirty eight (2) Using Commas Number Number using commas in Number using commas in Indian system of numeration International system of numeration (i) 637534 6,37,534 637,534 (ii) 86543 86,543 86,543 (iii) 27860038 2,78,60,038 27,860,038 (b) Arranged in ascending order, these numbers are as follows: 86543, 637534, 27860038 Arranged in descending order, these numbers are as follows: 27860038, 637534, 86543
10 MATHEMATICS VI (II) (a) (1) Using Placement Boxes S. No. Number T Cr Cr T La La T Th Th H T O (i) 748645 7 4 8 6 4 5 Number name Seven lakh forty eight thousand six hundred forty five (ii) 97654 9 7 6 5 4 Ninety seven thousand six hundred fifty four (iii) 38970049 3 8 9 7 0 0 4 9 Three crore, eighty nine lakh seventy thousand forty nine (2) Using Commas Number Number using commas in Indian Number using commas in system of numeration International system of numeration (i) 748645 7,48,645 748,645 (ii) 97654 97,654 97,654 (iii) 38970049 3,89,70,049 38,970,049 (b) Arranged in ascending order, these numbers are as follows: 97654, 748645, 38970049 Arranged in descending order, these numbers are as follows: 38970049, 748645, 97654 (III) (a) (1) Using Placement Boxes S. No. Number T Cr Cr T La La T Th Th H T O (i) 526423 5 2 6 4 2 3 Number name Five lakh twenty six thousand four hundred twenty three (ii) 75432 7 5 4 3 2 Seventy five thousand four hundred thirty two (iii) 16750027 1 6 7 5 0 0 2 7 One crore sixty seven lakh fifty thousand twenty seven (2) Using Commas Number Number using commas in Number using commas in Indian system of numeration International system of numeration (i) 526423 5,26,423 526,423 (ii) 75432 75,432 75,432 (iii) 16750027 1,67,50,027 16,750,027