IMPORTANT DEFINITIONS. Crores Lakhs Thousands Ones Period TC C TL L TTH TH H Tens Unit Place

Size: px

Start display at page:

Download "IMPORTANT DEFINITIONS. Crores Lakhs Thousands Ones Period TC C TL L TTH TH H Tens Unit Place"

Cordelia Payne
5 years ago
Views:

1 Chapter 1 Knowing our Numbers 1. Indian place value chart for 9-digit number. IMPORTANT DEFINITIONS Crores Lakhs Thousands Ones Period TC C TL L TTH TH H Tens Unit Place 2. International place value short for a 9 digit number. Millions Thousands Ones Period HM TH M HTH TTH TH H T U Place 1 million = 10 lakhs Ten million = 1 core Hundred million = 10 crores Comparing Numbers The arrangement of numbers from the smallest to the greatest is called ascending order. Ex: 2789, 3560, 4567, 7662, 7665 The arrangement of numbers from the greatest to the smallest is called descending order. Ex: 7665, 7662, 4567, 3560, 2789 If two numbers have an unequal number of digits, then the number with the greater number of digits is greater. If two numbers have an equal number of digits, then the number with the greater digit is greater. The greatest single-digit number is 9. When we add 1 to this single-digit number, we get 10, which is the smallest two-digit number. Therefore, the greatest single-digit number +1 = the smallest twodigits number. The greatest two digit -number is 99. When we add 1 to this two-digit number, we get 100, which is the smallest three- digits number. Therefore, the greatest two-digit number +1 = the smallest threedigits number. The greatest three- digits number is 999. When we add 1 to this three- digits number, we get 1000, which is the smallest four- digits number. Therefore, the greatest three- digits number +1 = the smallest four- digits number.

2 Class VI-Mathematicss 2 The greatest four- digits number is When we add 1 to this four-digit number, we get 10,000, which is the smallest five- digits number. Therefore, the greatest four-digit number +1 = the smallest five -digits number. The greatest five- digits number is When we add 1 to this five-digit number, we get 1,00,000, which is the smallest six digits number. Therefore, the greatest five -digits number +1 = the smallest six- digits number. The number, that is, one with five zeroes (100000), is called one lakh. Place value of a digit 7308 Place value of 7 is 7000 Place value of 7 is thousands Face value of a digit is the value of the digit itself face value of 7 is 7. Extended form of a number Ex: Use of Commas Commas in international system As per international numeration, the first comma is placed after the hundreds place. Commas are then placed after every three digits. Example: (i) 8,876,547. The number can be read as eight million eight hundred seventy-six thousand five hundred and fortyseven. (ii)56,789, 056 The number can be read as fifty-six million seven hundred eighty-nine thousand and fifty-six. Billions Millions Thousands Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Use the following place value chart to identify the digit in any place in the international system. Comparison of the Indian and the international numeration systems: Indian Numeration Crore Ten Lakh Lakh Ten Thousand Thousand Hundred Tens Ones Numbers International Numeration Units of measurement 1 metre=100 centimetres 1 kilogram = 1,000 grams 1 kilometre = 1000 metres 1 litre=1,000 millilitres Ten Million Million Hundred Thousand Ten Thousand Thousand Hundred Tens Ones

3 Class VI-Mathematicss 3 Estimation TIPS FOR COMPETITIVE LEVEL Before estimation, we must know how to round off a number to the nearest ten, nearest hundred, nearest, thousand, etc. For the same, we need the rules given below: Rounding a number to the Nearest Ten Step 1: Step 2: See the ones digit of the given number. If ones digit is less than 5, replace ones digit by 0, and keep the other digits as they are. Step 3: If ones digit is 5 or more, increase tens digit by 1, and replace ones digit by 0. Rounding a Number to the Nearest Hundred Step 1: Step 2: Step 3: See the tens digit of the given number. If tens digit is less than 5, replace each one of tens and ones digits by 0 and keep the other digits as they are. If this digit is 5 ore more, increase hundred digit by 1 and replace each digit on its right by 0. Rounding a Number to the Nearest Thousand Step 1: Step 2: Step 3: See the hundreds digit of the given number. If hundred digit is less than 5, replace each one of hundreds, tens and ones digit by 0 and keep the other digits as they are If hundreds digit is 5 or more, increase thousands digit by 1 and replace each digit on its right by 0. We may extent the ideas for larger numbers. Estimation of Number The estimation of a number is a reasonable guess of the actual value. Estimation means approximating a quantity to the accuracy required. This is done by rounding off the numbers involved and getting a quick, rough answer. The numbers 1, 2, 3 and 4 are nearer to 0. So, these numbers are rounded off to the lower ten. The numbers 6, 7, 8 and 9 are nearer to 10. So, these numbers are rounded off to the higher ten. The number 5 is equidistant from both 0 and 10, so it is rounded off to the higher ten. Eg: i) We round off 31 to the nearest ten as 30 ii) We round off 57 to the nearest ten as 60 iii) We round off 45 to the nearest ten as 50hy

4 Class VI-Mathematicss 4 The numbers 1 to 49 are closer to 0. So, these numbers are rounded off to the nearest hundred. The numbers 51 to 99 are closer to the lower hundred. So, these numbers are rounded off to the higher hundred. The number 50 is rounded off to the higher hundred. Eg: i) We round off 578 to the nearest 100 as 600. ii) We round off 310 to the nearest 100 as 300. Similarly, 1 to 499 are rounded off to the lower thousand, and 501 to 999 to the higher thousand. The number 500 is equidistant from both 0 and 1000, and so it is rounded off to the higher thousand. Eg: i) We round off 2574 to the nearest thousand as ii) We round off 7105 to the nearest thousand as Estimation of sum or difference: When we estimate sum or difference, we should have an idea of the place to which the rounding is needed. Examples i) Estimate We can say that > 4689 We shall round off the numbers to the nearest thousands is rounded off to is rounded off to 5000 Estimated sum: =24000 ii) Estimate We shall round off these numbers to the nearest hundreds is rounded off to is rounded off to 500 Estimated difference: =900 Estimation of the product: To estimate the product, round off each factor to its greatest place, then multiply the rounded off factors. Examples iii) Estimate The first number, 92, can be rounded off to the nearest ten as 90. The second number, 578, can be rounded off to the nearest hundred as 600. Hence, the estimated product = = 54,000 Estimating the outcome of number operations is useful in checking the answer. Use of Brackets

5 Class VI-Mathematicss 5 Using brackets: Brackets help in simplifying an expression that has more than one mathematical operation. If an expression that includes brackets is given, then turn everything inside the bracket into a single number, and then carry out the operation that lies outside. Example: 1. (6 + 8) 10 = = (8 + 3) (9 4) = 11 5 = 55 Expanding brackets: The use of brackets allows us to follow a certain procedure to expand the brackets systematically. Example: 1. 8 x 109 = 8 x ( ) = 8 x x 9 = = x 108 = ( ) x ( ) = ( ) x ( ) x8 8 = 100 x x x x 8 = = 1134 Roman Numerals Hindu Arabic number system: Many years ago, Hindus and Arabs developed a number system called the Hindu Arabic number system. It is the name given to the number system that we use today. Roman nnumerals: It is the numeral system that originated in ancient Rome. This numeral system is based on certain letters, which are given values and are used as numerals. The following are the seven number symbols used in the Roman numeral system, and their values: I V X L C D M Seven letters of English alphabet, i.e. I, V, X, L, C, D and M, are used to represent Roman numerals. Roman numerals do not have a symbol for zero. Roman numerals are read from left to right, and are arranged from the largest to the smallest. Multiplication, division and other complex operations were difficult to perform on Roman numerals. So Arabic numerals were used. The Roman numerals for the numbers 1-15 are shown below: I II III IV V VI VII VIII IX X XI XII XIII XIV XV We can find these roman numerals in some clocks. Rules for Roman numeralsnumerals 1. In Roman numerals, a symbol is not repeated more than thrice. If a symbol is repeated, its value is added as many times as it occurs.

6 Class VI-Mathematicss 6 For example, if the letter I is repeated thrice, then its value is three The symbols V, L and D are never repeated If a symbol of smaller value is written to the right of a symbol of greater value, then its value gets added to the symbol of greater value. For example, 3.4. iin case of VI, I is written to the right of V. It means that 1 should be added to 5. Hence, its value is 64. If a symbol of smaller value is written to the left of a symbol of greater value, then its value is subtracted from the symbol of greater value. For example, in case of IV, I is written to the left of V. It means that 1 should be subtracted from 5. Hence, its value is The symbols V, L and D are never written to the left of a symbol of greater value, so V, L and D are never subtracted. For example, we write 15 as XV and not VX. The symbol I can be subtracted from V and X only. For example, the value of IV is four and the value of VI is six. The symbol X can be subtracted from L, M and C only. For example, X is subtracted from L to arrive at 40, which is represented by XL PART I: MISCELLANEOUS DOMAIN 1. Fill in the blanks i) The smallest natural number is. ii) Thousands make a million. iii) The smallest six digit number using three different digits is iv) =. v) LXIV XIX =. 2. Tick the correct answer i) Predecessor of the predecessor of 101 is. a)100 (b) 99 (c) 98 (d) 97 ii) Five lakh fifty thousand and five in figures is. a)(a) 5, 50,505 (b) 5,55,500 (c) 5,55,005 (d) 5,55,050 iii) The difference of the place values of the two 9 s in is _ a)(a) 89,99,910 (b) 98,99,910 (c) 89,99,900 (d) 89,98,910 ivii) ( ) = ( ) a)(a) 244 (b) 356 (c) 400 (d) 500 iv) The reciprocal of 1 is.

7 Class VI-Mathematicss 7 a)(a) 0 (b) 1 (c) Do the following: Round off the number in the left side box as asked. 6,78,729 a) Nearest 10 = (d) 0 1 b) Nearest 100 = c) Nearest 1000 = d) Nearest 10,000 = 4. Determine the difference of the place values of two 7 s in Write the smallest three digits numbers which does not change if the digits are written in reverse order. 6. Find the difference between the number 279 and that obtained by reversing its digits. 7. Write the greatest 7-digit having three different digits. 8. Find the difference of the place value and the face value of the digits 2in Write all possible three digit numbers using the digits 6,0,4,when (1)repetition of the digits are not allowed (2) repetition of the digits are allowed 10. Determine the product of the place values of two 5 s in In a four digit number the digit at the thousand s place is 4 and the digit in the one,s place is twice that in the thousand s place. The numbers has no hundreds. The ten s place digit is the difference between the digits in the thousand s place and the hundred s place.find the number Which digit have the same place value and the face value in A certain nine digit number has only ones in ones period, only two s in the thousand period and only threes in millions period.write this number in words in the Indian system Write the numbers in according to Indian or international system of numeration in words a) 19,15,60,050 b) 20,48,74,873 c) 97,97,97,997 d) 507,476,123 e) 100,005, Write the numbers in according to Indian or international system of numeration in numerals a) Five crores, five lakhs, five thousands b) Fifty six crores, nineteen lakh, eight thousand and fifteen. c) One hundred ten million, two hundred thousand and three hundred forty eight d) Two hundred thirty four million, eighteen thousand and three hundred forty eight e) One million, one thousand and one f) One crore, one lakh, one thousand, one hundred and one 156. Write each of the following in the Expanded notation form a) 8,40,32,167 b) 90,909,090 c) 17,865,432

8 Class VI-Mathematicss 8 d) 6,05,43,091 e) 10,00, Write the Expanded notation form in numerals for the following a) 1,00,00, ,00, ,00, , , b) 4 10,00, , , c) 90,00, , d) 8 10,00, How many numbers exist between 51 and 100? 19. How many thousands make a million? 20. How many millions make a billion? 21. How many lakhs make a million? 22. How many millions make ten crores? Fill in the gap: a) 1 crore= lakhs b) 1 Millions= thousands c) 10 crores= millions d) 1 Hundred thousand= lakhs 24. In a town there are men,39842 women and children. Find the total population of town? 25. Raghav deposited Rs.14572, Rs.20450, Rs.2987 and 9852 in the bank in four different months. Find the total deposit in the bank. 26. A shopkeeper gives a discount of Rs on an article which costs Rs.50.what is the selling price of the article? 27. Roni went to the market with Rs.1000.She bought a saree for Rs.375, a pair of shoes for Rs.155 and a shawl for Rs.2.85.How much money is left with him? 28. The monthly fee for a student in a school is Rs.310.If there are 620 students in the school, find the total monthly collection of fees. 29. Gita has Rs.78,592 with her. She placed an order for purchasing 39 radio sets at Rs.1234 each. How much money will remain with her after the purchase? 30. There are fifteen students in a row. If there are 737 students, calculate the number of rows in which they stand? 31. Find the largest six digits number which is divisible by 120 exactly

9 Class VI-Mathematicss The distance between the park and house of a student is 1Km 575m.Everyday he walks both ways between the park and his house. Find the total distance covered by him in a week s time? 33. Find the sum of the greatest and the least numbers that can be written using the digits 9,8,0,6,4 each only once Round off each of the following to the nearest A. Tens I. 12,096 II. 808 III. 28,295 IV. 84 B. Hundreds I. 7,289 II. 8,074 III. 14,627 IV. 28,826 C. Thousands I. 7,832 II. 9,567 III. 4,36,952 IV. 9, List all the numbers which will be rounded off to the nearest ten as Find the greatest and the smallest numbers which will be rounded off to the nearest hundreds as Give a rough estimate (by rounding off to the nearest hundreds) and a closer estimate (by rounding off to nearest tens) a) ,318 b) 1,89,768-1,45,890 c) ,904 d) Estimate the product using a general rule a) b) c) d) 67, Write the following in Roman Numerals a) 89 b) 996 c) 175 d) 1015 e) 197 f) 2359 g) 236 h) i) 478 j) k) l) Write the following in Hindu-Arabic numerals a) CCXLII b) CCXXIX c) CDLXXVIII d) XCIX e) DCXIV f) LXXXIX g) DXCVIII h) KCIX i) CDXLVI j) DCXCVII Which of the following are meaningless? a) VX b) KKKCCXI c) IC d) VC e) CI f) IL g) IXX h) XXXX

10 Class VI-Mathematicss 10 i) XLIX j) LXXXIX HIGHER ORDER THINKING SKILLS (HOTS) 26. In a town there are men,39842 women and children. Find the total population of town? 27. Raghav deposited `14572, `20450, `2987 and 9852 in the bank in four different months. Find the total deposit in the bank. 28. A shopkeeper gives a discount of Rs on an article which costs Rs.50.what is the selling price of the article? 29. Roni went to the market with `1000.She bought a saree for `375, a pair of shoes for `155 and a shawl for `2.85.How much money is left with him? 30. The monthly fee for a student in a school is `310.If there are 620 students in the school, find the total monthly collection of fees. 31. Gita has `78,592 with her. She placed an order for purchasing 39 radio sets at `1234 each. How much money will remain with her after the purchase? 32. There are fifteen students in a row. If there are 737 students, calculate the number of rows in which they stand? 33. Find the largest six digits number which is divisible by 120 exactly 34. The distance between the park and house of a student is 1Km 575m. Everyday he walks both ways between the park and his house. Find the total distance covered by him in a week s time? 35. Find the sum of the greatest and the least numbers that can be written using the digits 9,8,0,6,4 each only once. PART II: MULTIPLE CHOICE QUESTIONS 1. In a five digit number, the digit in the hundred s place is 2 and the digit in the one s place is twice the digit in the hundred s place. The number has no thousands. The digit in the ten-thousands place is the sum of the digit in the hundred s place and the digit in the one s place. The digit in the ten s place is the digit in the ten-thousands place minus 1. The number is (a) (b) (c) (d) If x and y are negative, then which of the following statements is/are always true? I. x + y is positive II. xy is positive III. x y is positive (a) I only (b) II only (c) III only (d) I and III only 3. A number x, when divided by 7, leaves a remainder 1 and another number y, when divided by 7 leaves the remainder 2. What will be the remainder if x + y is divided by 7? (a) 2404 (b) 4808 (c) 3648 (d) If * means adding 6 times the second number to the first number, then (1 * 2) * 3 equals. (a) 121 (b) 31 (c) 93 (d) 91

11 Class VI-Mathematicss What least number should be subtracted from so that the resulting number is divisible by 3 but not by 9? (a) 1 (b) 3 (c) 4 (d) 7 6. A student was asked to find the sum of all the prime numbers between 10 and 40. He found the sum as 180. Which of the following statements is true? (a) He missed one prime number between 10 and 20. (b) He missed one prime number between 20 and 30. (c) He added one extra prime number between 10 and 20. (d) None of these 7. If the digit 1 is placed after a two digit number whose ten s digit is t and unit s digit is u, the new number is: (a) 10t + u+ 1 (b) 100t + 10u + 1 (c) 1000 t + 10u + 1 (d) t + u Find the remainder when is divided by25. (a) 0 (b) 2 (c) 4 (d) 6 9. Find the least values of x and y so that the number 5x 423y is divisible by 88. (a) 8, 2 (b) 7, 3 (c) 9, 4 (d) 6, The sum of three consecutive odd numbers is always divisible by I. 1 II. 3 III. 5 IV. 6 (a) Only I (b) Only II (c) Only I and III (d) II and IV ANSWERS 1. (b) 2. (b) 3. (c) 4. (b) 5. (c) 6. (d) 7. (b) 8. (b) 9. (a) 10. (b)

(i) Indian system (ii) International system. EXERCISE 1.1

(i) Indian system (ii) International system. EXERCISE 1.1 Knowing our Numbers Learn and Remember 1. Digit: In order to represent any number, we use ten symbols: 0, 1,2,3,4,5,6, 7,8,9. These ten symbols are called digits or figures. 2. Numeral: A group of digits