POINTS FOR QUICK REVISION

Transcription

1 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 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 = 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 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 , 4972, 18, 59785, 750 Sol is the greatest and 18 is the smallest. Q , 89423, 100, 5000, 310 Sol is the greatest and 100 is the smallest. Q , 75284, 111, 2333, 450 Sol is the greatest and 111 is the smallest. Q , 7691, 9999, 12002, 124 Sol 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, (c) 25286, 25245, 25270, (d) 6895, 23787, 24569, Make five more problems of this kind and give to your friends to solve. Sol. (a) The greatest number is The smallest number is (b) The greatest number is

3 KNOWING OUR NUMBERS 3 The smallest number is (c) The greatest number is The smallest number is (d) The greatest number is The smallest number is Five more problems of this kind are given below. 1. Find the greatest and the smallest numbers. 5936, 5079, 5788, 5672 Greatest : 5936 Smallest : Find the greatest and the smallest numbers , 31729, 31649, Greatest : Smallest : Find the greatest and the smallest numbers , 42367, 42329, Greatest : Smallest : Find the greatest and the smallest numbers , 31897, 31899, Greatest : Smallest : Find the greatest and the smallest numbers , 20791, 3987, 3978 Greatest : 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 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 Smallest (Note, the number cannot begin with the digit 0. Why?) (b) Digit 4 is always at tens place Greatest 4 Smallest 4

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 Smallest (c) Greatest Smallest (d) Greatest Smallest 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 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 ` 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, Sol. (a) The given numbers in ascending order 571, 847, 8320, 9754 (b) The given numbers in ascending order are : 9801, 25751, 36501, Q. 2. Arrange the following numbers in descending order. (a) 5000, 7500, 85400, 7861 (b) 1971, 45321, 88715, 92547

5 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 , 9865, 9431, 682 Ex , 8643, 7210, 460 Ex , 7532, 6101, 350 Ex , 6421, 5012, 240 Ex , 5310, 4001, 130 Sol. The numbers arranged in ascending order are as follows: , 958, 9431, , 736, 7210, , 625, 6101, , 514, 5012, , 403, 4001, 5310 Arrange the following numbers in descending order: Ex , 8600, 86500, 8972 Ex , 6400, 74300, 6750 Ex , 56432, 99826, Ex , 34210, 66604, Ex , 23101, 55513, Sol. The numbers arranged in descending order are as follows: , 8972, 8600, , 7000, 6750, , 93658, 56432, , 66604, 34210, , 55513, 23101, 1650 CHAPTER 1 (Page 6) Q. 1. Read and expand the numbers wherever there are blanks. Number Number Name Expansion Twenty thousand Twenty six thousand Thirty eight thousand four hundred Sixty five thousand seven hundred forty Eighty nine thousand, three hundred twenty four

6 6 MATHEMATICS VI Sol Fifty thousand Forty one thousand Forty seven thousand three hundred Fifty seven thousand six hundred thirty Twenty nine thousand four hundred eighty five Twenty nine thousand eighty five Twenty thousand eighty five Twenty thousand five Q. 2. Write five more 5-digit numbers, read them and expand them. Sol. Number Number Name Expansion Thirty thousand 3 10, Fifty two thousand 5 10, , Thirty six thousand 3 10, ,000 four hundred Sixty eight thousand 6 10, ,000 nine hundred forty Twenty four thousand 2 10, ,000 five hundred seventy nine (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 3,53, 500 Three lakh fifty three thousand 3 1,00, , 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, , nine hundred twenty eight ,07,928 Four lakh seven thousand nine 4 1,00, hundred twenty eight ,00,829 Four lakh eight hundred twenty nine 4 1,00, ,00,029 Four lakh twenty nine 4 1,00,

7 KNOWING OUR NUMBERS 7 Q. 1. What is 10 1 =? 2. What is =? 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 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 = = ,000 1 = 9, ,00,000 1 = 99, ,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) (ii) (iii) (iv) (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) CHAPTER 1 Number name Four lakh, seventy five thousand, three hundred twenty Expansion 4 1,00, , (Contd.)

8 8 MATHEMATICS VI (ii) Ninety eight lakh, forty seven thousand, two hundred fifteen 9 10,00, ,00, , (iii) Nine crore, seventy six lakh, forty-five thousand, three hundred ten 9 1,00,00, ,00, ,00, , , (iv) Three crore, four lakh, fifty eight thousand ninety four 3 1,00,00, ,00, , , (a) The smallest number is (b) The greatest number is (c) The numbers arranged in ascending order are as follows : , , , The numbers arranged in descending order are as follows : , , , Q. 2. Read these numbers (i) (ii) (iii) (iv) (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) Five lakh twenty seven thousand eight hundred sixty four (ii) Ninety five thousand four hundred thirty two (iii) One crore eighty nine lakh fifty thousand forty nine (iv) Seven crore two thousand five hundred nine

9 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) ,27, ,864 (ii) ,432 95,432 (iii) ,89,50,049 18,950,049 (iv) ,00,02,509 70,002,509 (b) Arranged in ascending order, these numbers are as follows : 95432, , , Arranged in descending order, these numbers are as follows : , , , 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 , 86543, II , 97654, III , 75432, (I) (a) (1) Using Placement Boxes CHAPTER 1 S. No. Number T Cr Cr T La La T Th Th H T O (i) Number name Six lakh thirty seven thousand five hundred thirty four (ii) Eighty six thousand five hundred forty three (iii) 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) ,37, ,534 (ii) ,543 86,543 (iii) ,78,60,038 27,860,038 (b) Arranged in ascending order, these numbers are as follows: 86543, , Arranged in descending order, these numbers are as follows: , , 86543

10 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) Number name Seven lakh forty eight thousand six hundred forty five (ii) Ninety seven thousand six hundred fifty four (iii) 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) ,48, ,645 (ii) ,654 97,654 (iii) ,89,70,049 38,970,049 (b) Arranged in ascending order, these numbers are as follows: 97654, , Arranged in descending order, these numbers are as follows: , , (III) (a) (1) Using Placement Boxes S. No. Number T Cr Cr T La La T Th Th H T O (i) Number name Five lakh twenty six thousand four hundred twenty three (ii) Seventy five thousand four hundred thirty two (iii) 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) ,26, ,423 (ii) ,432 75,432 (iii) ,67,50,027 16,750,027