CBSE Class 6 Mathematics Whole Numbers Assignment Set D

CBSE Assignment on Whole Numbers. Prepared by HOD Mathematics of one of the best schools in Delhi

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 addition is called the closure property of addition. For ex. : \( 3 + 4 = 7 \); \( 2 + 8 = 10 \) i.e., whole number + whole number = whole number.
• Commutative Property: If a and b are two whole numbers \( a + b = b + a \). This property of addition, where the order of addition does not alter the sum, is called the commutative property of addition. For ex. : \( 3 + 4 = 7 \). Also, \( 4 + 3 = 7 \), i.e., \( 3 + 4 = 4 + 3 \).
• Associative Property: If a, b and c are three whole numbers then, \( a + (b + c) = (a + b) + c \). In 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 ex. : \( 2 + (3 + 4) = (2 + 3) + 4 \Rightarrow 2 + 7 = 5 + 4 \Rightarrow 9 = 9 \).
• Additive Identity: 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 ex. : \( 7 + 0 = 7 = 0 + 7 \).

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, then the answer will not be a whole number. Hence subtraction is not closed under whole numbers. For ex. : \( 7 - 2 = 5 \) 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 ex. : \( a - b \neq b - a \); \( 7 - 2 \neq 2 - 7 \).
• 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 does not hold true for subtraction of whole numbers. For ex. \( (12 - 4) - 3 = 8 - 3 = 5 \); \( 12 - (4 - 3) = 12 - 1 = 11 \). \(\therefore (12 - 4) - 3 \neq 12 - (4 - 3) \).
• 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 ex. : \( 3 - 0 = 3 \).

PROPERTIES OF MULTIPLICATION

• Closure Property: If a and b are whole numbers, then their product \( a \times b = c \) will always be a whole number. That is whole numbers are closed under multiplication. For ex. : \( 7 \times 3 = 21, 6 \times 8 = 48, 3 \times 0 = 0 \).
• Commutative Property: In general \( a \times b = b \times a \) for all whole numbers a and b. Consider the following example \( 2 \times 3 = 3 \times 2 = 6 \); \( 8 \times 9 = 9 \times 8 = 72 \).
• Associative Property: If a, b and c are whole numbers, then \( (a \times b) \times c = a \times (b \times c) \). That is, whole numbers have the associative property of multiplication. For ex. : \( (3 \times 4) \times 2 = 3 \times (4 \times 2) \Rightarrow 12 \times 2 = 3 \times 8 \Rightarrow 24 = 24 \).
• Multiplicative Identity: \( 1 \times a = a \times 1 = a \). Hence, 1 is called the multiplicative identity for whole numbers. For ex. : \( 10 \times 1 = 1 \times 10 = 10 \); \( 3 \times 1 = 1 \times 3 = 3 \); \( 672 \times 1 = 1 \times 672 = 672 \); \( 0 \times 1 = 1 \times 0 = 0 \).
• Property of Zero: When any whole number a is multiplied by zero, the product is zero. That is, \( a \times 0 = 0 \times a = 0 \). For ex. : \( 27 \times 0 = 0 \times 27 = 0 \).

PROPERTIES OF DIVISION

• Closure Property: If a and b are whole number, then the quotient \( a \div b \) need not always be a whole number. So, division in whole numbers is not closed. For ex. : \( 6 \div 3 = 2 \), \( 6 \div 4 = 1\frac{1}{2} \), \( 6 \div 7 = \frac{6}{7} \). \( 1\frac{1}{2} \) and \( \frac{6}{7} \) are not whole numbers.
• Commutative Property: If a and b are whole numbers, then \( a \div b \) is not equal to \( b \div a \). So, the commutative property does not hold true for whole numbers. For ex. : \( 6 \div 3 = 2 \) is not the same as \( 3 \div 6 = \frac{1}{2} \).
• Associative Property: If a, b and c are whole numbers, then \( (a \div b) \div c \) is not equal to \( a \div (b \div c) \). \( (a \div b) \div c \neq a \div (b \div c) \). For ex. : \( (81 \div 9) \div 3 = 3 \) and \( 81 \div (9 \div 3) = 27 \). So, \( (81 \div 9) \div 3 \) is not equal to \( 81 \div (9 \div 3) \). 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 ex. : \( 6 \div 1 = 6, 8 \div 1 = 8 \). Hence, \( a \div 1 = a \).
2. If zero is divided by any whole number, the result will always be zero. For ex. : \( 0 \div 3 = 0 \). So, \( 0 \div a = 0 \).
3. Division of a whole number by zero is meaningless and is not allowed. For example, to speak of dividing 12 oranges between zero students is meaningless.

DISTRIBUTIVE PROPERTY

If a, b, c are whole numbers, then \( a \times (b + c) = a \times b + a \times c \). This property is called the distributive property of multiplication over addition. For ex. : \( 7 \times (8 + 3) = 7 \times 8 + 7 \times 3 \Rightarrow 7 \times 11 = 56 + 21 \Rightarrow 77 = 77 \).

If a, b, c are whole numbers \( (b > c) \), then \( a \times (b - c) = a \times b - a \times c \). This property is called the distributive property of multiplication over subtraction. For ex. : \( 5 \times (7 - 3) = 5 \times 7 - 5 \times 3 \Rightarrow 5 \times 4 = 35 - 15 \Rightarrow 20 = 20 \).

Question. Solve using distributive property: (i) \( 8 \times 107 \) (ii) \( 18 \times 95 \)
Answer: (i) \( 8 \times 107 = 8 \times (100 + 7) = 8 \times 100 + 8 \times 7 = 800 + 56 = 856 \)
(ii) \( 18 \times 95 = 18 \times (100 - 5) = 18 \times 100 - 18 \times 5 = 1800 - 90 = 1710 \)

Question. Using suitable arrangement, find the product of : (i) \( 8, 9, 25, 3 \) (ii) \( 4, 897, 25 \) (iii) \( 250, 2986, 4 \) (iv) \( 4000, 625, 32 \) (v) \( 125, 40, 8, 25 \)
Answer: (i) \( (8 \times 25) \times (9 \times 3) = 200 \times 27 = 5400 \)
(ii) \( (4 \times 25) \times 897 = 100 \times 897 = 89700 \)
(iii) \( (250 \times 4) \times 2986 = 1000 \times 2986 = 2986000 \)
(iv) \( 4000 \times 625 \times 16 \times 2 = (4000 \times 2) \times (625 \times 16) = 8000 \times 10000 = 8,00,00,000 \)
(v) \( (125 \times 40) \times (8 \times 25) = 5000 \times 200 = 10,00,000 \)

Question. Find the value of the following using properties of multiplication: \( 37 \times 865 + 18 \times 865 - 49 \times 865 - 6 \times 865 \)
Answer: \( 865 \times (37 + 18 - 49 - 6) = 865 \times (55 - 55) = 865 \times 0 = 0 \).

Question. 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?
Answer: Cost of 25 pencils = \( 25 \times \text{Rs. } 2 = \text{Rs. } 50 \)
Cost of 25 rulers = \( 25 \times \text{Rs. } 8 = \text{Rs. } 200 \)
Total cost = \( 50 + 200 = \text{Rs. } 250 \)
Alternatively, \( 25 \times (2 + 8) = 25 \times 10 = \text{Rs. } 250 \).

EXERCISE 

Question. 1. How many times does the digit 2 occur between 1 and 100 ?
(a) 10
(b) 9
(c) 12
(d) 20
Answer: D

Question. 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) \( a \times b \)
(d) \( 2a + b \)
Answer: B

Question. 3. A student wrote \( 5 + 24 + 25 + 6 = 5 + 25 + 24 + 6 \). Which property of addition did he use?
(a) Closure property
(b) Commutative property
(c) Associative property
(d) Property of zero
Answer: B

Question. 4. \( 27 + 52 + 73 + 10 = 100 + \square \). Which value shall come in the box?
(a) 52
(b) 73
(c) 62
(d) 37
Answer: C

Question. 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) \( 4 + (9 + 23) = (4 + 9) + 23 \)
Answer: C

Question. 6. The population of a village is 1500. If 489 are men and 472 are women, find the number of children.
(a) 549
(b) 439
(c) 559
(d) 539
Answer: D

Question. 7. The value of \( 300 \times 4 \times 0 \times 10 \) is
(a) 1200
(b) 12000
(c) 120000
(d) 0
Answer: D

Question. 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
Answer: C

Question. 9. Which of the following will not represent zero?
(a) \( 1 \div 0 \)
(b) \( 0 \times 0 \)
(c) \( \frac{0}{2} \)
(d) \( \frac{10 - 10}{2} \)
Answer: A

Question. 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) \( a \times (b + c) = (a \times b) + (a \times c) \)
(c) \( a \div (b \div c) = (a \div b) \div c \)
(d) \( (a \times b) \times c = a \times (b \times c) \)
Answer: C

Question. 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
Answer: B

Question. 12. How many whole numbers are smaller than 9 ?
(a) 1
(b) 2
(c) 3
(d) 9
Answer: D

Question. 13. The smallest whole number is :
(a) 0
(b) 9
(c) 2
(d) 1
Answer: A

Question. 14. The predecessor of whole number is :
(a) 2
(b) 9
(c) 0
(d) Does not exist
Answer: C

Question. 15. The additive identity of whole number 1 is :
(a) 0
(b) 1
(c) 2
(d) none of these
Answer: A

Question. 16. In whole numbers \( a - b \square b - a \), \( \square \) means :
(a) \( = \)
(b) \( > \)
(c) \( < \)
(d) \( \neq \)
Answer: D

Question. 17. The multiplicative identity of whole number is :
(a) 0
(b) 1
(c) 9
(d) none of these
Answer: B

Question. 18. Which is not defined ?
(a) \( 4 \div 2 \)
(b) \( 0 \div 4 \)
(c) \( 9 \div 3 \)
(d) \( 3 \div 0 \)
Answer: D

Question. 19. The relation \( a + b = b + a \), where a, b are whole number is :
(a) closed
(b) associative
(c) commutative
(d) none of these
Answer: C

Question. 20. Subtraction in whole numbers is :
(a) commutative
(b) closed
(c) associative
(d) none of these
Answer: D

Question. 21. Which is not the successor of any whole number ?
(a) 1
(b) 0
(c) 2
(d) 9
Answer: B

Question. 22. The predecessor of 9099 is :
(a) 9088
(b) 9098
(c) 9100
(d) 9091
Answer: B

Question. 23. The whole number which is not a natural number, is :
(a) 1
(b) 0
(c) 9
(d) 2
Answer: B

Question. 24. \( a + a = 1 \), for which whole number it is not true?
(a) 1
(b) 2
(c) 0
(d) none of these
Answer: C

Question. 25. Which of the following does not give whole number?
(a) \( 12 \div 4 \)
(b) \( 1 \div 8 \)
(c) \( 0 \div 2 \)
(d) none of these
Answer: B

Question. 26. Which number are closed under the operation :
(a) addition
(b) subtraction
(c) multiplication
(d) addition and multiplication
Answer: D

EXERCISE - 

Question. 1. We know that \( 0 + 0 = 0 \). ls there some other whole number p such that \( p + p = p \)?
Answer: No, zero is the only number

Question. 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 ?
Answer: Both travels equal distance (320 km)

Question. 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?
Answer: Both sold equal number of tickets. (50 tickets)

Question. 4. Show that \( 7 \times (12 \times 15) = (7 \times 12) \times 15 \)
Answer: \( 7 \times 180 = 84 \times 15 \Rightarrow 1260 = 1260 \). (This demonstrates the Associative Property of Multiplication).

Question. 5. Solve using distributive property: (i) \( 12 \times 197 \) (ii) \( 37 \times 102 \)
Answer: (i) \( 12 \times (200 - 3) = 2400 - 36 = 2364 \)
(ii) \( 37 \times (100 + 2) = 3700 + 74 = 3774 \)

Question. 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) \( 7 \times (32 \times 56) = (7 \times 32) \times \) _______ .
(iv) ________ is the additive identity for whole number.
Answer: (i) Whole Number (ii) sum, same, commutative property (iii) 56 (iv) 0 (zero)

Question. 7. The population of a village is 10725. 1 out of every 15 persons is uneducated. How many educated persons live in the village?
Answer: 10010

Question. 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?
Answer: 147

Question. 9. What number should replace each n ?
(i) \( 3 (n + 6) = (3 \times 5) + (3 \times 6) \)
(ii) \( (7 \times 4) + (n \times 3) = 7 (4 + 3) \)
(iii) \( (9 \times 8) + (8 \times 8) = (9 + 8) n \)
Answer: (i) n = 5 (ii) n = 7 (iii) n = 8

Question. 10. Find each of the following products by using properties of multiplication :
(i) \( 972 \times 8 + 972 \times 2 \)
(ii) \( 46 \times 982 + 27 \times 982 - 58 \times 982 - 15 \times 982 \)
(iii) \( 957 \times 10 \times 583 - 483 \times 9570 \)
Answer: (i) 9720 (ii) 0 (iii) 957000

Question. 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?
Answer: Rs. 1160

Question. 12. Find the value of each of the following :
(i) \( (3278 \div 3278) - (5098 \div 5098) \)
(ii) \( 0 \div 975 \)
(iii) \( 701 - (1869 \div 1869) \)
Answer: (i) 0 (ii) 0 (iii) 700

Question. 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
Answer: (i) True (ii) False (iii) False (iv) True (v) False

Question. 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.
Answer: 204

Question. 15. How many whole numbers are there between 3 and 23 ?
Answer: 19

Question. 16. How many whole numbers, each less than 47, are there in Hindu-Arabic system of numeration ?
Answer: 47

Question. 17. The digits 6 and 9 of the number 36490 are interchanged. Find the difference between the original number and the new number.
Answer: 2970

Question. 18. Using most convenient combinations, find the sum \( 1802 + 2652 + 3376 + 1024 + 2348 + 98 \).
Answer: 11300

Question. 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.
Answer: 444

Question. 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?
Answer: Tuesday

Question. 21. In each of the following fill in the blanks, so that the statement is true :
(a) \( (500 + 7) \times (300 - 1) = 299 \times \) ________
(b) \( 888 + 777 + 555 = 111 \times \) _________
(c) \( 75 \times 425 = (70 + 5) \times (25 + \) _______)
(d) \( 89 \times (100 - 2) = 98 \times (100 - \) ________)
(e) \( 9 \times (10000 + \) ________) \( = 98766 \)
Answer: (a) 507 (b) 20 (c) 400 (d) 11 (e) 974

Question. 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 \( a = b \).
Answer: (a) False (b) False (c) False (d) True (e) True