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NCERT 6TH CLASS MATHEMATICS CHAPTER WHOLE NUMBERS

    NATURAL NUMBERS

When we count a set of objects, we start counting from one then go on to two, three, four, etc.
This is the natural way of counting any set of objects.
Hence, 1, 2, 3, 4,... are called counting numbers or natural numbers. The number of students in a class, the number of days in a week, and the number of trees in a garden are all examples of natural numbers.
 

   ZERO

Let us consider an example to understand the concept of zero. If we want to divide 7 sweets equally among 3 children, 1 sweet will be left. But if we were to divide 6 sweets equally among 3 children, we are left with no sweets.
Zero means absence of the item (or no item)
 

   WHOLE NUMBERS

The numbers 1, 2, 3,...., are called natural number or counting numbers. Let us add one more number, i.e., zero (0), to the collection of natural numbers. Now the numbers are 0, 1, 2, 3,.... . These numbers are called whole numbers.
 

  PROPERTIES OF ADDITION

· Closure Property : If ‘a’ and ‘b’ are two whole numbers and their sum is c, i.e., a + b = c, then c will always be a whole number. This property of a addition is called the closure property of addition.
For Example. : 3 + 4 = 7
2 + 8 = 10 i.e., whole number + whole number = whole number
 
· Commutative Property : If a and b are two whole numbers a + b = a. This property of addition, where the order of addition does not alter the sum, is called the commutative property of addition.
For Example. : 3 + 4 = 7
Also, 4 + 3 = 7
i.e., 3 + 4 = 4 + 3
 
· Associative Property : If a, b and c three whole numbers then, then a + (b + c) = (a + b) + c. 
ln other words, in the addition of whole numbers, the sum does not change even if the grouping is changed. This property is called the associative property of addition.
For Example. : 2 + (3 + 4) = (2 + 3) + 4
2 + 7 = 5 + 4
9 = 9
 
· Additive ldentity : If a is a whole number, then a + 0 = 0 + a = a.
Hence, zero is called the additive identity of the whole numbers because it maintains (or does not change) the identity (value) of the numbers during the operation of addition.
For Example. : 7 + 0 = 7 = 0 + 7
 

   PROPERTIES OF SUBTRACTION

· Closure Property : If a and b are two whole numbers, then a – b will be a whole number only if a is greater than b or a is equal to b. If a is smaller than b, than the answer will not be a whole number. Hence subtraction is not closed under whole numbers.
For Example. : 7 – 2 = 5 is whole number
but 3 – 8 is not a whole number
 
· Commutative Property : If a and b are two distinct whole numbers, then a – b is not equal to b – a. Hence, the commutative property is not true for subtraction of whole numbers.
For Example. : a – b ¹ b – a
7 – 2 ¹ 2 – 7
 
· Associative Property : If a, b and c are whole numbers, then (a – b) – c is not equal to a – (b – c). So, the associative property also not hold true for subtraction of whole numbers
For Example. (12 – 4) – 3 = 8 – 3 = 5
12 – (4
...
3) = 12 – 1 = 11
\ (12 – 4) – 3 ¹ 12 – (4 – 3)
 
· Property of Zero : If zero is subtracted from any whole number, then the result is the number itself. a – 0 = a, for any whole number a.
For Example. : 3 – 0 = 3.
 

   PROPERTIES OF MULTIPLICATION

· Closure Property : If a and b are whole numbers, then their product a × b = c will always be a whole number. That is whole numbers are closed under multiplication.
For Example. : 7 × 3 = 21, 6 × 8 = 48, 3 × 0 = 0
·
Commutative Property : In general a × b = b × a for all whole numbers a and b.
Consider the following example
2 × 3 = 3 × 2 = 6
8 × 9 = 9 × 8 = 72
 
· Associative Property : If a, b and c are whole numbers, then
(a × b) × c = a × (b × c)
That is, whole numbers have the associative property of multiplication.
For Example. : (3 × 4) × 2 = 3 × (4 × 2)
12 × 2 = 3 × 8 
24 = 24
 
· Multiplicative ldentity : 1 × a = a × 1 = a. Hence, 1 is called the multiplicative identity for whole numbers.
For Example. : 10 × 1 = 1 × 10 = 10
3 × 1 = 1 × 3 = 3
672 × 1 = 1 × 672 = 672
0 × 1 = 1 × 0 = 0
 

· Property of Zero : When any whole number a is multiplied by zero, the product is zero. That is,

a × 0 = 0 × a = 0
For Example. : 27 × 0 = 0 × 27 = 0
 

   PROPERTIES OF DIVISION

 
 Closure Property : If a and b are whole number, then the quotient a + b need not always be a whole number. So, division in whole numbers is not closed.
For Example. : 6÷3 = 2, 6÷ 4 = 1 , 6÷ 7 = 
1 and  are not whole numbers.
· Commutative Property : If a and b are whole numbers, then a÷ b is not equal to b÷ a. So, the commutative property does not hold true for whole numbers.
For Example. : 6÷ 3 = 2 is not the same as 3÷ 6 = 
 
· Associative Property : If a, b and c are whole numbers, then (a÷ b)÷ c is not equal to a÷ (b÷ c). (a÷ b)÷ c ¹ a÷ (b÷ c)
For Example. : (81÷ 9)÷ 3 = 3 and 81÷ (9÷ 3) = 27
So, (81÷ 9)÷ 3 is not equal to 81÷ (9÷ 3) 
Hence, the associative property does not apply to the division of whole numbers.
 

   SPECIAL PROPERTIES

1. Whenever a whole number is divided by 1, we get the same whole number as the answer.
For Example. : 6÷ 1 = 6, 8÷ 1 = 8
If 6 sweets are divided between 2 children, we have 6÷ 2 = 3. Each child gets 3 sweets. If 6 sweets are divided among 3 children, then 6÷ 3 = 2. Each child gets 2 sweets in this case. If 6 sweets are given to one child, then 6÷ 1 = 6. The child gets 6 sweets. So, when we divide by taking 1 as the divisor, the quotient (answer) is the same as the dividend.
Hence, a÷ 1 = a
2. If zero is divided by any whole number, the result will always be zero.
For Example. : 0÷ 3 = 0
If there are zero chocolates or no chocolate in a packet and we divide into equal parts, each part will still have only zero chocolates.
So, 0÷ a = 0
3. Division of a whole number by zero is meaningless and is not allowed.
For Exampleample, to speak of dividing 12 oranges between zero students is meaningless.
 

   DISTRIBUTIVE PROPERTY

If a, b, c are whole numbers, then a × (b + c) = a × b + a × c
This property is called the distributive property of multiplication over addition.
For Example. : 7 × (8 + 3) = 7 × 8 + 7 × 3
7 × 11 = 56 + 21
77 = 77
If a, b, c are whole numbers (b > c), then a × (b – c) = a × b – a × c
This property is called the distributive property of multiplication over subtraction.
For Example. : 5 × (7 – 3) = 5 × 7 – 5 × 3
5 × 4 = 35 – 15
20 = 20
Example. Solve using distributive property.
 
(i) 8 × 107 (ii) 18 × 95
Sol. (i) 8 × 107 = 8 × (100 + 7)
= 8 × 100 + 8 × 7 = 800 + 56 = 856
 
(ii) 18 × 95 = 18 × (100 – 5)
= 18 × 100 – 18 × 5 = 1800 – 90 = 1710
 
Example. Using suitable arrangment, find the product of :
(i) 8, 9, 25, 3               (ii) 4, 897, 25                   (iii) 250, 2986, 4
(iv) 4000, 625, 32 (v) 125, 40, 8, 25
Sol. Since the number may be grouped in any order we group those numbers which make the calculations most conveneint.
 
(i) 
(ii) 4 × 897 × 25 = (4 × 25) × 897 = 100 × 897 = 89700
(iii) 250 × 4 × 2986 = 250 × 4 × 2986
= (250 × 4) × 2986 = 1000 × 2986
= 2986000.
 
(iv) 4000 × 625 × 32
= 4000 × 625 × 16 × 2
= (4000 × 2) × (625 × 16)
= 8000 × 10000 = 8,00,00,000.
 
(v) 125 × 40 × 8 × 25
= (125 × 40) × (8 × 25) = 5000 × 200 = 10,00,000
 
Example. Find the value of the following using properties of multiplication.
37 × 865 + 18 × 865 – 49 × 865 – 6 × 865
Sol. 37 × 865 + 18 × 865 – 49 × 865 – 6 × 865
= 865 × (37 + 18 – 49 – 6)
= 865 × (55 – 55) = 865 × 0 = 0.
Example. 25 sets containing a pencil and a ruler are made. The cost of each pencil is Rs. 2 and that of a ruler is Rs. 8. What is the total cost of 25 sets?
Sol. Cost of 25 pencils = 25 × Rs. 2 = Rs. 50
Cost of 25 rulers = 25 × Rs. 8 = Rs. 200
\ Total cost = 25 × 25 + 25 × 8
Alternatively,
 
Total cost = 25 × Total cost of pencil and ruler
= 25 × Rs. (2 + 8) = 25 × Rs. 10 = Rs. 250
Thus, we can see here,
25 × (2 + 8) = 25 × 2 + 25 × 8 = 250, i.e.,
we have used distributive property.
 

   OBJECTIVE TYPE

 
Q.1 How many times does the digit 2 occur between 1 and 100 ?
(A) 10                   (B) 9                       (C) 12              (D) 20
 
Q.2 Given two whole number a and b, which of the following may not always be whole numbers.
(A) a + b              (B) a – b                  (C) a × b          (D) 2a + b
 
Q.3 A student wrote
5 + 24 + 25 + 6 = 5 + 25 + 24 + 6.
 
Which property of addition did he use?
(A) Closure property
(B) Communtative property
(C) Associative property
(D) Property of zero
 
Q.4 27 + 52 + 73 + 10 = 100 + . Which value shall come in the box?
(A) 52                     (B) 73                     (C) 62                  (D) 37
 
Q.5 Which of the following statements does not represent a property of addition of whole number?
(A) 38 + 53 = 53 + 38
(B) 16 + 7 is a whole number
(C) 899 + 10 = 8990
(D) 4 + (9 + 23) = (4 + 9) + 23
 
Q.6 The population of a village is 1500. If 489 are men and 472 are women, find the numebr of children.
(A) 549                    (B) 439                     (C) 559                 (D) 539
 
Q.7 The value of 300 × 4 × 0 × 10 is
(A) 1200                 (B) 12000                  (C) 120000           (D) 0
 
Q.8 Find the number of pages in a book which has on an average 305 words on a page, and contains 2,32,715 words altogether?
(A) 1111 pages           (B) 1001 pages              (C) 763 pages              (D) 973 pages
 
Q.9 Which of the following will not represent zero?
(A) 1÷ 0                     (B) 0 × 0                         (C)                              (D) 
Q.10 Which of the following statements is not true for three whole numbers a, b and c ?
(A) a + (b + c) = (a + b) + c                              (B) a × (b + c) = (a × b) + (a × c)
(C) a÷ (b÷ c) = (a÷ b)÷ c                                  (D) (a × b) × c = a × (b × c)
 
Q.11 On dividing a number by 68, we get 269 as quotient and 0 as remainder. On dividing the same number by 67, what will be the remainder?
(A) 0                (B) 1                       (C) 2                   (D) 3
 
Q.12 How many whole numbers are smaller than 9 ?
(A) 1                (B) 2                       (C) 3                   (D) 9
 
Q.13 The smallest whole number is :
(A) 0                (B) 9                       (C) 2                   (D) 1
 

   OBJECTIVE TYPE

 
Q.1 How many times does the digit 2 occur between 1 and 100 ?
(A) 10                       (B) 9                      (C) 12                   (D) 20
 
Q.2 Given two whole number a and b, which of the following may not always be whole numbers.
      (A) a + b             (B) a – b                 (C) a × b               (D) 2a + b
 
Q.3 A student wrote
       5 + 24 + 25 + 6 = 5 + 25 + 24 + 6.
       Which property of addition did he use?
     (A) Closure property
     (B) Communtative property
     (C) Associative property
     (D) Property of zero
 
Q.4 27 + 52 + 73 + 10 = 100 + . Which value shall come in the box?
      (A) 52                (B) 73                     (C) 62                        (D) 37
 
Q.5 Which of the following statements does not represent a property of addition of whole number?
     (A) 38 + 53 = 53 + 38
     (B) 16 + 7 is a whole number
     (C) 899 + 10 = 8990
     (D) 4 + (9 + 23) = (4 + 9) + 23
     
Q.6 The population of a village is 1500. If 489 are men and 472 are women, find the numebr of children.
       (A) 549                 (B) 439                        (C) 559                        (D) 539
 
Q.7 The value of 300 × 4 × 0 × 10 is
       (A) 1200               (B) 12000                    (C) 120000                  (D) 0
 
Q.8 Find the number of pages in a book which has on an average 305 words on a page, and contains 2,32,715 words altogether?
      (A) 1111 pages       (B) 1001 pages           (C) 763 pages               (D) 973 pages
 
Q.9 Which of the following will not represent zero?
      (A) 1÷ 0                  (B) 0 × 0                    (C)                                 (D) 
 
Q.10 Which of the following statements is not true for three whole numbers a, b and c ?
       (A) a + (b + c) = (a + b) + c                      (B) a × (b + c) = (a × b) + (a × c)
       (C) a÷ (b÷ c) = (a÷ b)÷ c                           (D) (a × b) × c = a × (b × c)
 
Q.11 On dividing a number by 68, we get 269 as quotient and 0 as remainder. On dividing the same number by 67, what will be the remainder?
        (A) 0                    (B) 1                       (C) 2                 (D) 3
 
Q.12 How many whole numbers are smaller than 9 ?
       (A) 1                     (B) 2                       (C) 3                 (D) 9
 
Q.13 The smallest whole number is :
        (A) 0                    (B) 9                      (C) 2                  (D) 1
 
Q.14 The predecessor of whole number is :
        (A) 2                    (B) 9                      (C) 0                  (D) Does not exist
 
Q.15 The additive identity of whole number 1 is :
        (A) 0                    (B) 1                      (C) 2                  (D) none of these
 
Q.16 In whole numbers a – b o b – a, o means :
        (A) = (B) > (C) < (D) ¹
 
Q.17 The multiplicative identity of whole number is :
       (A) 0                  (B) 1               (C) 9                      (D) none of these
 
Q.18 Which is not defined ?
        (A) 4÷ 2            (B) 0÷ 4         (C) 9÷ 3                 (D) 3÷ 0
 
Q.19 The relation a + b = b + a, where a, b are whole number is :
        (A) closed                      (B) associative                   (C) commutative                   (D) none of these
 
Q.20 Subtraction in whole numbers is :
        (A) commutative           (B) closed                           (C) associative                      (D) none of these
 
Q.21 Which is not the successor of any whole number ?
        (A) 1                       (B) 0                                (C) 2                      (D) 9
 
Q.22 The predeccessor of 9099 is :
        (A) 9088                 (B) 9098                          (C) 9100                (D) 9091
 
Q.23 The whole number which is not a natural number, is :
        (A) 1                       (B) 0                                 (C) 9                     (D) 2
 
Q.24 a + a = 1, for which whole number it is not true?
       (A) 1                        (B) 2                                 (C) 0                     (D) none of these
 
Q.25 Which of the following does not give whole number?
       (A) 12÷ 4                 (B) 1÷ 8                           (C) 0÷ 2                 (D) none of these
 
Q.26 Which number are closed under the operation :
      (A) addition              (B) subtraction                 (C) multiplication                 (D) addition and multiplication
 

   SUBJECTIVE TYPE

 
Q.1 We know that 0 + 0 = 0. ls there some other whole number p such that p + p = p
 
Q.2 Ali cycle for 16 days, riding 20 km each day. Sam cycles 20 days, riding 16 km each day. Who cycles a further distance ?
 
Q.3 Tripti sold 5 books of raffle tickets. Hari sold 10 books of raffle tickets. If the books sold by
       Tripti had 10 tickets each, and those sold by Hari had 5 tickets each, who sold more tickets?
 
Q.4 Sow that 7 × (12 × 15) = (7 × 12) × 15
 
Q.5 Solve using distributive property
      (i) 12 × 197                (ii) 37 × 102
 
Q.6 Fill in the blanks
     (i) If any two whole number are added, we always get a ________ number.
     (ii) If any two whole number a and b are added, a to b or b to a, the ________ is always _______. 
           This property is called ________ property of addition of whole number.
     (iii) 7 × (32 × 56) = (7 × 32) × _______ .
     (iv) ________ is the additive identity for whole number.
 
Q.7 The population of a village is 10725. 1 out of every 15 persons is uneducated. How many educated persons live in the village?
 
Q.8 Sheela brought a Hindi novel from the library which had 378 pages. She read 152 pages on the first two days. If she read 79 pages on the third day, how many pages remain unread?
 
Q.9 What number should replace each n ?
     (i) 3 (n + 6) = (3 × 5) + (3 × 6)
     (ii) (7 × 4) + (n × 3) = 7 (4 + 3)
    (iii) (9 × 8) + (8 × 8) = (9 + 8) n
 
Q.10 Find each of the following products by using properties of multiplication :
     (i) 972 × 8 + 972 × 2
    (ii) 46 × 982 + 27 × 982 – 58 × 982 – 15 × 982
    (iii) 957 × 10 × 583 – 483 × 9570
  
Q.11 Ashok buys 20 notebook and 20 pens. The cost of each notebook is Rs.45 and that of each pen is 
        Rs. 13. Find the amount of money he spent?
 
Q.12 Find the value of each of the following :
        (i) (3278÷ 3278) – (5098÷ 5098)
       (ii) 0÷ 975
      (iii) 701 – (1869÷ 1869)
 
Q.13 State whether the following statements are true or false
       (i) Zero is the smallest whole number
      (ii) Every whole number is greater than zero
      (iii) 64 – 36 = 36 – 64
      (iv) 75 + 0 = 75
       (v) 1 is the additive identity for Whole number
 
Q.14 Find the sum of the four numbers given below :
        Successor of 32, predecessor of 49
        predecessor of the predecessor of 56 and successor of the successor of 67
 
Q.15 How many whole numbers are there between 3 and 23 ?
 
Q.16 How many whole numbers, each less than 47, are there in Hindu-Arabic system of numeration ?
 
Q.17 The digits 6 and 9 of the number 36490 are interchanged. Find the difference between the original number and the new number.
 
Q.18 Using most convenient combinations, find the sum 1802 + 2652 + 3376 + 1024 + 2348 + 98.
 
Q.19 There are 222 red balls in a basket. A boy takes out 6 red ball from it and replace them by 12 white balls. He continues
         to do so till all red balls are replaced by white balls. Determine the number of white balls put in the basket.
 
Q.20 The first February of a leap year falls on a FRIDAY. On what day of the week would the first April of the year fall?
 
Q.21 In each of the following fill in the blanks, so that the statement is true :
        (a) (500 + 7) × (300 – 1) = 299 × ________
        (b) 888 + 777 + 555 = 111 × _________
        (c) 75 × 425 = (70 + 5) × (25 + _______)
        (d) 89 × (100 – 2) = 98 × (100 – ________)
        (e) 9 × (10000 + ________) = 98766
 
Q.22 Which of the following statement are true and which are false ?
       (a) On a number line, every whole number represents exactly one point and every point is represented by exactly one whole number.
       (b) Every whole number is the successor of another whole number.
       (c) The largest 4–digit number formed by the digits 6, 7, 0, 9 using each digit only once is 9706.
       (d) The sum of two whole number is always greater than or equal to their difference.
       (e) If a and b are two whole numbers such that a – b = b – a, then a = b.