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NCERT 6TH CLASS MATHEMATICS CHAPTER KNOWING OUR NUMBERS

 

PREDECESSOR-SUCCESSOR

        Predecessor is 1 less than the givOF A NUMBERen number. For Ex. : Predecessor of 59028 is 59027.

        Successor is 1 more than the given number. Successor of 9999 is 10000.

COMPARISON OF NUMBERS:

     (a) Greater number has more number of digits.
     (b) To compare two numbers having same number of digits, start comparing from the leftmost position.

  If the leftmost digits are equal, move to the next digits.

  Ex. Which is greater of 270346 and 48356?

  Sol. 270346 has 6 digits

  48356 has 5 digits
  6 digits are more than 5 digits

  270346 is greater than 48356

  Greater number has more number of digits.

  Ex. Find the greatest and the smallest numbers from the following group of numbers :

  23787, 6895, 24569, 24659

  Sol. Greatest number : 24659

  Smallest number : 6895

ASCENDING AND DESCENDING ORDER

Ascending order -

      arrangement of numbers from smallest to largest

Descending order -

      arrangement of numbers from largest to smallest

 Ex. Arrange the following number in ascending order :

 257536, 38952, 385081, 365062

 Sol. The smallest number is 38952. Other numbers greater than 38952, in order are 257536, 365062 and 385081.

 The numbers in ascending order are :

 38952, 257536, 365062, 385081

  Ex. Arrange the following numbers in descending order :

  19710, 887151, 453212, 925473

  Sol. The greatest number is 925473. Other numbers smaller than 925473 in order are 887151, 453212 and 19710.

 The numbers in descending order are :

  925473, 887151, 453212, 19710

USE OF COMMAS

     While reading and writing large numbers, it creates confusion as there are many digits in a number. To avoid making mistakes,

     we use commas to divide the number into different periods. We can divide a number into different periods by using either the lndian

     system of numeration or the lnternational system of numeration. But first let us learn about place value.

PLACE VALUE

    The basis of the number system is place value. It is this place value which gives value to the number.

 Definition :

    Face value of a digit in a numeral is the value of the digit itself at whatever place it may be.

    Place value of a digit in a given number is the value of the digit because of the place or the position of the digit in the number.

    Place-value of a digit = Face-value of the digit × value of the place

Place value and Face Value :

    Every digit has two values —

   the place value and the face value. The face value of a digit does not change while its place value changes according to its position and number.

 

    Number 

     Digit   

        Face Value       

           Place Value        

   53,694

      5

           5 

            50,000

 

      3

            3

             3,000

 

      6

            6

             600

 

      9

           9

              90

 

      4

           4

               4

   

Expanded form of a Number :

    If we express a given number as the sum of its place value, it is called its expanded form.

    Ex. Express

    (i) 3,64,029
    (ii) 2,75,00,386 in expanded form.

    Sol. Place value of 3 = 3 ×

...
 Place value of 6 = 6 × 10000
   Place value of 4 = 4 × 1000
   Place value of 0 = 0 × 100
   Place value of 2 = 2 × 10
   Place value of 9 = 9 × 1

  The expanded from of 3,64,029 is 3 × 100000 + 6 × 10000 + 4 × 1000 + 0 × 100 + 2 × 100 + 9 × 1

INTERNATIONAL SYSTEM OF NUMBERS

 

                                      Billions                                                                   Milllions                                                                              Thousands

   Hundred Billion

    Ten Billion

    One Billion

  Hundred Million

     Ten Million

        One Million     

   Hundred Thousad     

    Ten Thousand    

   One Thousand    

100, 000,000, 000

    12 Digits

10,000,000,000

    11Digits

   1,000,000,000

   10 Digits

100,000,000

   9 Digits

10,000,000

 8 Digits

     1,000,000

       7 Digit

          100,000

          6 Digits

   1 0,000

    5 Digits

    1,000

   4 Digits

 

Reading and Writing of Numbers :

    In lndian System of numbers, we divide the given number into periods starting from the right.

   The first period called units period consists of 3 digits while each of the next periods called thousands

   period, lakhs period, crores period respectively consists of 2 digits.

   Each period is separated by a comma (,).
   ln lnternational System of Numbers we make groups of 3 digits starting from right and separate each group by using a comma (,).
   Now consider the six digit number 764325. In the lndian system, the number 7 is in the place of lakh and therefore the number is

   named as seven lakh sixty-four thousand three hundred twenty-five. lt is written as 7,64,325. In the lnternational system,

   this numebr is named as seven hundred sixty-four thousand three hundred twenty-five and is written as 764,325.   

   he first three digits form the right-hand side are considered as ones, i.e., 325 ones and the next three digits to the

  left of the digit 3 are considered as thousands, i.e. 764 thousands.

  Ex. Write the name of the numbers according to lndian system of numeration.

  (a) 1275834                 (b) 30870209

  Sol. (a) 12,75,834 = Twelve lakh seventy five thousand eight hundred thirty four

  (b) 3,08,70,209 = Three crore eight lakh seventy thousand two hundred nine

   Ex. Write the name of the number according to lnternational system of numeration.

   (a) 7452283                (b) 48049831               (c) 699985102

   Sol.  (a) 7452283 = 7,452,283 = Seven million four hundred fifty two thousand two hundred thirty three.

   (b) 48049831 = 48,049,831 = Forty eight million forty nine thousand eight hundred thirty one.

   (c) 699985102 = 699,985,102 = Six hundred ninety nine million nine hundred eighty five thousand one
   hundred two.

MAKING NUMBERS

   (a) Making number without repetition of digits :

   In case of non-repetition of digits, it is better if we start making the number from left.

   Ex. Write the greatest and the smallest 5-digit numbers by using each of digits 8, 4, 7, 0, 2 only one.

   Sol. For the greatest number, we write the greatest digit 8 in the T-thousands column. Next smaller digit in the thousands column and so on.

       The greatest number = 87420.

      T - T h          

       T h           

           H           

           T           

               O                 

         8

        7

          4

          2

              0

   For the smallest number, we write the smallest digit in the T-thousands column. But here 0 is the smallest digit. 0 is not written

   on the extreme left of a number. So, we write 2 in the T-thousands column and 0 in the thousands column, Next digit greater than

    2 is written in the hundreds column and so on.

   The smallest number = 20478

         T - T h         

           T h          

           H           

          T              

         O            

            2

           0

           4

          7

         8

    Ex. Make the greatest and the smallest 5-digit numbers using any five different digit with 4 in the tens place.

    Sol. First of all write 4 in the tens column. For the greatest number, we write the greatest digit 9 in the

    T - thousands column, next smaller digit in the thousands column so on.

          T- Th             

           Th         

            H         

          T           

         O              

           9

            8

            7

         4

         6

    The greatest number = 98746

    For the smallest number also, write 4 in the tens column first of all. Then write 0 in the thousands column.

    Write 1 in the T-thousands column. Next greater digit in the hundreds column and so on.
    The smallest number = 10243

           T- Th           

         Th          

             H           

       T           

            O            

              1

          0

             2

       4

            3

   (b) Making number with repetition of digits : ln case of repetition of digit, it is better if we start making number from right.

   Ex. Write the greatest and smallest numbers of 4 digits using all the digits 8,0,5.

   Sol. For greatest number, select the smallest digit 0 and write in the ones column, Next greater digit is written in the tens column.

   Next greater digits 8 is written in the hundreds column. Since no digit greater than 8 given, so we repeat 8 in the thousands column.

  T- Th           

         H          

             T          

 

            O            

       8

         8

             5

 

            3

 

 

Greatest Number

   The greatest number = 8850

   For smallest number, select the greatest digit 8 and write in the ones column.

   Next smaller digit in tens column and so on. Repeat the smallest digit in the end.

   But here 5 is smaller than 8 and then comes 0 which cannot be repeated in the end.
   So, we write 5 in the end and repeat 0 in the tens place.

           Th               

             H            

            T               

                 O             

           5

             0

            0

                8

Smallest Number

   The smallest number = 5008

NOTE :

   i. For greatest number, greatest digit is repeated.

  ii. For smallest number, smallest digit is repeated.

ESTIMATION IN NUMBER OPERATIONS

   You must have come across news headlines involving large numbers. For example, ‘50,000 people participated

   in the marathon.’ ‘5 lakh people gathered for a dip in the Ganges.’
   We also see and read news about disasters, strikes, bandhas, etc.

   For example, ‘80 crore rupees lost due to the fire.’
   ‘9 lakh people evacuated.’
   ‘The loss due to bandh is 1 crore.’ 
   The numbers mentioned above do not represent the exact number of people or rupees.

   They are only speaking of a nearest value.
   Rounding up off numbers is an approximation. This means that when we speak of 50,000 people,

   what we really mean is the crowd was between 45,000 and 55,000.
   This rounding up helps us to get aproximate answers in addition, subtraction,

   multiplication and division. Suppose Rs.3847 and Rs.8348 are the costs of a DVD player and a T.V., respectively.

   If a friend asked you how much you paid for these two items, you need not to add the correct value of each.

   You can say that it costs you around Rs.12,000. When we are cooking for a party, the approximate number of people

   expected for the party, and the approximate quantity of each item required per person are considered. Similarly, the quantity

  of things to be purchased for a party, the amount of cement to be purchased to construct a wall, the amount of paint to be bought

  to paint a house are all approximations.

Estimating (Rounding) to the Nearest Ten :

   To round off a number to the nearest ten consider the ones digit.

   If the ones digit is 5 or greater than 5, then change the ten’s digit to the next higher digit and ones digit to zero.

   If the ones digit is less than 5, then leave the tens digit unchanged but change the ones digit to zero.

Estimating (Rounding) to the Nearest Hundred :

   To round off a number to the nearest to the nearest hundred,

   consider the tens digit. If the ten’s digit is 5 or greater than 5, then change the hundreds digit to the next higher digit

   and tens, ones digits to zeros. If the tens digit is less than 5, then leave the hundreds digit unchanged but change the tens and ones digits to zeros.

Estimating (Rounding) to the nearest Thousand :

   To round off a number to the nearest thousand, consider the hundreds digit.

   If this digit is 5 or greater than 5, then change the thousands digit to the next higher digit are change all the other digits before that to zeros.

   If the hundreds digit is less than 5, then leave the thousands digit unchanged but change all the other digits before that to zero.

Estimation in Sums or Differencess :

   Ex. Estimate and compare with the actual sum

   (a) 730 + 998 (b) 12904 + 2888

   Sol. (a) We round off to the nearest hundred

   730 is rounded off to 700
   998 is rounded off to 1000
   Estimated sum = 1700
   700+1000=1700

   Actual sum = 730 + 998 = 1728

   730+998=1728

   Estimation is quite reasonable,


  (b) We round off to the nearest tousand 13000
  12904 is rounded off to 13000
  2888 is rounded off to 3000
  Estimation is quite reasonble
  Actual sum = 12904 + 2888 = 15792
  Estimation is quite reasonable

  12904+2888=15792

  Ex. Give a rough estimate and also a close estimate of 439 + 334 + 4317

  Sol. Rough estimate : 400 + 300 + 4000 = 4700

  For closer estimate, we round off to the nearest hundred
  439 is rounded off to 400 400
  334 is rounded off to 300
  4317 is rounded off to 4300
  Closer estimated sum = 5000

  Ex. Estimate 8325 – 491

  Sol. If we round off to thousand, we get

  8325 rounds off to 8000
  491 rounded off to 0
  Estimated difference = 8000 – 0 = 8000

  This does not give a reasonable difference, so we round off to the nearest hundred.
  8325 rounds to 8300
  491 round to 500
  Estimated difference = 7800
  This is reasonable estimation.
  8300–500=7800

  Estimation in Products : While estimating in products, we adopt the following rules :

  (i) Round each factor to its greatest place, in other words, if a factor contains 2 digits,

      round it off to the nearest ten and if it contains 3 digits, then round it off to the nearest hundred and so on.
  (ii) Do not round off any 1-digit factor.

      Ex. Estimate the following products :

  (a) 87 × 313 (b) 9 × 795 (c) 898 × 785

  Sol. (a) 87 is rounded off to 90

   313 is rounded off to 300

    Estimated product = 90 × 300 = 27000

  (b) 9 is not rounded off [ it is a one-digit no.]
      795 is rounded off to 800

   Estimated product = 9 × 800 = 7200 (c) 898 is rounded off to 900

  785 is rounded off to 800

  Estimated product = 900 × 800 = 720000

Estimation in Quotients :

   In the process of estimation in quotients, we round off the divided and the divisor before the process of division.

   Ex. Estimate the following quotients :

  (a) 81 ÷ 17 (b) 7525 ÷  365

      Sol. (a) 81 is rounded to 80

   17 is rounded to 20


   To get the estimated quotient think of dividing 80 by 20 or 8 by 2.
   Estimated quotient = 8/2= 4

  (b) 7525 is rounded to 8000
   365 is rounded to 400
   To get the estimated quotient think of dividing 80 by 4.

 Estimated quotient = 20

Use of number in everyday life

   Numbers are used immensly in our everyday life, such as measuring the length of a small object as pencil,

   the distance between two given places ; the weight of an orange, the weight of a ship,

   the amount of juice in a glass and the amount of water in a like.

   Small lengths are measured in millimeter (mm) and centimetre (cm) while bigger

   lengths are measured in metre (m) and kilometre (km).
   Metre (m) is the standard unit of length and we define it as
   1 m = 100 cm = 1000 mm

   1 cm = 10mm

   100 cm = 100 × 10 = 1000 mm

   1 km = 1000 m
   Also, 1 km = (1000m × 1000 mm) = 1000000 mm
  Similarly, the units of weight are as under
   1 gm = 1000 mg
   1 kg = 1000gm

   1 kg = (1000 gms × 1000) mg = 1000000 mg

    For capacity or volume,
   1 L = 1000 mL and 1 kL = 1000 L 
   1 kL = 1000 L × 1000 mL = 100000 mL 

   Ex. A tin of biscuits has 14 kg of biscuits. Express the weight in milligrams.

   Sol. Since 1 kg = 1000 gm and 1 gm = 1000 mg

   1 kg = (1000 × 1000) mg = 1000000 mg

   14 kg = 14 × 1000000 mg = 14000000 mg

   Ex. The population of Rajasthan is 5,64,73,122, of Goa is 13,43,998 and of Karnataka is 5,27,33,958.

   What is the combined population of the three states.

   Sol. Population of Rajasthan = 5,64,73,122

   Population of Karnataka = 5,27,33,958. Pupulation Goa = 13,43,998

   Total population of three states=

   5,64,73,122 + 13,43,998 + 5,27,33,958 = 11,05,51,078 i.e. Eleven crore five lakh fifty one thousand seventy eight.

   Ex. What must be added to 34,52,629 to make it equal to 6 corore.

  Sol. 6 crore = 6,00,00,000

   required number = 6,00,00,000 – 34,52,629 = 5,65,47,371

  Ex. There are 785 students on roll in a residential public school. If the annual fee per student is Rs.62,606.

  What is the total fee collected annually by the school.

  Sol. Annual fee of one student = Rs.62,606

  Number of student = 785

  Total annual collection of fee

     = Rs. 62,606 × 785
     = Rs. 4,91,45,710

   Ex. Find the number of pages in a book which has on an average 207 words on a page, and contains

   201411 words altogether.
   Sol. Number of pages = 201411÷  207 = 973
   Thus, the number of pages in the book = 973

 USE OF BRACKETS

    Raju bought 6 pencils from the market each of Rs. 2. 
    His brother Ramu also bought 8 pencils of the same type. Raju and Ramu both calculated the total cost but in their own ways.

    Raju found that they both spent Rs.28 and he used the following method :
    Rs. 6 × 2 + Rs. 8 × 2
     = Rs. 12 + Rs. 16
     = Rs. 28 
    But Ramu found an easier way. He did 6 + 8 = 14 and then Rs. 2 × 14 = Rs.28. The use of brackets makes this sum easy.

    It can be done as follows : 
    Rs. 2 × (6 + 8) 
    = Rs. 2 × 14
    = Rs. 28 
    Clearly we first solve the operation inside the bracket and then multiply it by the number outside.

ROMAN NUMERALS

   One of the earlist systems of writing numerals is the Roman Numeral system. This system is still in use in many places.

   For example, some faces of clocks show hours in Roman numerals; we use Roman numerals to write numered list; etc.

   Unlike the Hindu-Arabic numeral system, Roman numeral system uses seven basic symbols to represent different numbers. The symbols are as follows :
   l = 1, V = 5, X = 10, L = 50, C = 100
  D = 500, M = 1000

RULES TO FORM ROMAN NUMERALS

  We can form different Roman numerals using the symbols and the following rules.

Rule-1

   If a symbol is repeated one after the other, its value is added as many times as it occurs. For example lll = 1 + 1 + 1 = 3

  XX = 10 + 10 = 20

Rule-2

The symbols l, X, C and M can be repeated up to a maximum of three times. For example l = 1,

   ll = 2, lll = 3
   X = 10, XX = 20, XXX = 30, C = 100
   CC = 200, CCC = 300, M = 1000, MM = 2000,
   MMM = 3000

Rule-3

   The symbols V, L and D (i.e., 5, 50 and 500 respectively) can never be repeated in a Roman numeral,

Rule-4

  If a symbol with a smaller value is written on the right of a symbol with a greater value, then its value is added to the value of the greater symbol. For example

   Xll = 10 + 2 = 12, LX = 50 + 10 = 60
   DCCCX = 500 + 300 + 10 = 810

Rule-5

   If a symbol with a smaller value is written on the left of a symbol with a larger value, then its value is subtracted from the value of the greater symbol.

   For example,
   lV = 5 – 1 = 4, lX = 10 – 1 = 9, CD = 500 – 100 = 400, Vl = 5 + 1 = 6, Xl = 10 + 1 = 11, DC = 500 + 100
   = 600

  Ex. Write the Roman Numerals

  (a) 105 (b) 213

  Sol. (a) 105 = 100 + 5 = CV

  (b) 213 = 200 + 10 + 3 = CCXlll

  Ex. Write in Hindu Arabic numerals :

  (a) CXXXV (b) CXLl

  Sol. (a) CXXXV = C + XXX + V = 100 + 30 + 5 = 135 (b) CXLl = C + XL + l = 100 + 40 + 1 = 141

 

OBJECTIVE TYPE

  Q.1 Which of the number shown below is meaningless ?

  (A) Vlll             (B) XX             (C) XVl              (D) VXXX

  Q.2 Find the difference in the place value of the digit 5 in the number 57568.

  (A) 49500          (B) 45900         (C) 49000         (D) 49050

  Q.3 What is the difference of 1 million and 900 tens?

  (A) 100900        (B) 991000        (C) 91000         (D) 919000

  Q.4 How much is 50 less than by 1 million ?

  (A) 9950            (B) 99950           (C) 999950         (D) 9999950

  Q.5 Which of the following numbers when rounded off to the nearest ten thousand gives 500000?

  (A) 492811         (B) 495213         (C) 589200           (D) 513076

  Q.6 Which one of the following is the best estimation of 5663 × 2234?

  (A) 5000 × 2000           (B) 5700 × 2200              (C) 5660 × 2230              (D) 5660 × 2300

  Q.7 Using the digits 1, 4, 6 and 8 each only once, how many four digit odd numbers can be formed?

  (A) 4              (B) 5                (C) 3                 (D) 6

  Q.8 In the product of 37 and 23, what is the place value of the digit which is prime?

  (A) 500              (B) 700               (C) 800          (D) 50

  Q.9 How many numbers of 3-digit are formed by using the digits 0, 1 and 2?

  (A) 5            (B) 6               (C) 10             (D) 4

  Q.10 l as a Roman numeral, am CMXClX. Break me up and then can you recognise me?

  (A) 9910               (B) 999              (C) 1109               (D) 1119

  Q.11 Which one of the following is the smallest numeral?

  (A) 15673          (B) 15700                 (C) 15198               (D) 15623

  Q.12 The largest number using each of digits 5, 7, 8, 9 is :

  (A) 9875                (B) 5879                 (C) 8759               (D) 7589

  Q.13 The successor of 49,999 is :

  (A) 49,998            (B) 50,000             (C) 49,990            (D) 49,000

  Q.14 The smallest 4-digit number using 2, 0, 9, 5 is :

  (A) 9520             (B) 0295             (C) 2059              (D) 5209

  Q.15 Which digit is at thousands place in 57, 168 :

  (A) 6                (B) 7               (C) 1               (D) 8

  Q.16 The place value of 9 in 7690453 is :

  (A) 900           (B) 9000         (C) 90000        (D) 90

   Q.17 789500 comes just after :

  (A) 789400             (B) 789501             (C) 789499            (D) 789498

  Q.18 1 quintal = ......... kg

   (A) 10         (B) 100             (C) 1000           (D) 100000

   Q.19 The successor of the greatest 4-digit number is

   (A) 9999                  (B) 9998                (C) 9909         (D) 10000

   Q.20 LXV can be written in Hindu Arabic numaral as :

   (A) 55               (B) 60                  (C) 65               (D) 70

 

SUBJECTIVE TYPE

 

  Q.1 Write the Predecessor of 7000?

  Q.2 Which is greater 72389 and 72391?

  Q.3 Arrange the following in ascending order:

    2345, 2543, 3452, 4325, 2435

  Q.4 Write the following numbers as numerals.

  (a) Sixty two lakh forty five thousand six hundred thirty five
  (b) Nine crore fifty eight lakh sixty one thousand eighty nine

  Q.5 Write the following numbers in expanded notation.

  (a) 2,84,231             (b) 52,11,568                   (c) 6,04,18,517           (d) 8,91,81,213

  Q.6 According to the 1991 census, the number of people who spoke the following languages were : Assamese : 13079696

  Hindi : 337272114
  Konkani : 1760607
  (a) Write the above numbers according to the lndian and lnternational system of numeration.

  (b) Write the above numbers in words according to the lndian system of numeration.

  (c) Write the above numbers in words according to the lnternational system.

  Q.7 Write the greatest and the smallest 5 digits numbers by using each of the digits 3, 5, 7, 0, 9 only once?

  Q.8 Write the greatest and smallest numbers of 4 digits using all the digits 7, 0, 6?

  Q.9 Give the approximate value by estimating.

  (a) 3228 + 572    (b) 8010 – 2507       (c) 32 × 58          (d) 108 × 47

  Q.10 Write the equivalent Roman numeral of each of the following Hindi-Arabic numeral.

  (a) 46     (b) 90          (c) 120         (d) 150

  Q.11 Write the equilvalent Hindu Arabic numerals of the following Roman numerals.

  (a) XXl          (b) CCX      (c) DCC          (d) LXlll

  Q.12 12 drums of milk have 84 litres of milk in them. Find the capacity of one drum in millilitres.

  Q.13 The number of candidates appearing for class 10 board examination conducted by CBSE was 14, 58,

  937 in year 2002; 16, 93, 487 in year 2003; 24, 13, 468 in year 2004 and 40, 05, 093 in year 2005.

  Find the total number of candidates who appeared for the examination in these four years.

  Q.14 The number of scoters produced in a year was 25, 43, 163. Out of these 16, 43, 078 were sold. How many were still left?

  Q.15 A milk depot sells 657 litres of milk every day. How much milk will it sell in 1 year?

  Q.16 The students of class Vl of a school collected Rs. 3, 37, 875 for Prime Minister’s Relief fund.

  If each child contributed Rs.255, how many children are there in the school?

  Q.17 Write all possible three digit numbers (without repeating the digits), by using the digits.

  (i) 6, 7, 5            (ii) 9, 0, 2

  Q.18 Use the given digits without repetition and make the smallest and the greatest four digit numbers.

  (i) 2, 1, 5, 6        (ii) 7, 8, 0, 9            (iii) 4, 6, 3, 5           (iv) 8, 3, 2, 4            (v) 2, 5, 9, 0          (vi) 1, 9, 6, 3

  Q.19 Make the greatest and the smallest four digit numbers by using any one digit twice :

  (i)6, 3, 2              (ii) 1, 0, 6                (iii) 7, 9, 4                (iv) 2, 5, 0

  Q.20 Make the greatest and the smallest 4-digit numbers using any four different digits, with the condition given below:

  (i) Digit 6 is always in thousands place

  (ii) Digit 4 is always in hundreds place

  (iii) Digit 7 is always in tens place
  (iv) Digit 1 is always in ones place
  (v) Digit 9 is always in thousands place

 (vi) Digit 0 is always in hundreds place

  (vii) Digit 5 is always in tens place
  (viii) Digit 3 is always in ones place