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N = {1, 2, 3, ...} 0 + × W = {0, 1, ...} —

Mathematics · Class 6 · STBB

Unit 2 — Whole Numbers

Sindh Textbook Board — Solved notes with verified answers

Key Concepts ✓  Exercise 2.1 ✓  Exercise 2.2 ✓  Exercise 2.3 ✓  Exercise 2.4 ✓  Review Exercise 2 ✓

1. Key Concepts — Whole Numbers

• Natural numbers N = {1, 2, 3, 4, ...} — used for counting. 1 is the smallest natural number.
• Whole numbers W = {0, 1, 2, 3, 4, ...} — natural numbers together with 0. 0 is the smallest whole number.
• The only difference between N and W is the number “0” — 0 belongs to W but not to N.
• Predecessor = a number minus 1 (the number just before it). Successor = a number plus 1 (the number just after it). E.g. predecessor of 1 is 0; successor of 1 is 2.
• Number line: a line on which whole numbers are marked at equal distances. A number is bigger than any number on its left and smaller than any number on its right.

Laws of Whole Numbers

• Commutative law (addition): a + b = b + a   e.g. 9 + 30 = 30 + 9
• Associative law (addition): (a + b) + c = a + (b + c)   e.g. (3 + 7) + 12 = 3 + (7 + 12) = 22
• Additive identity: 0 — a + 0 = 0 + a = a
• Commutative law (multiplication): a × b = b × a   e.g. 5 × 6 = 6 × 5 = 30
• Associative law (multiplication): (a × b) × c = a × (b × c)   e.g. (3 × 2) × 4 = 3 × (2 × 4) = 24
• Multiplicative identity: 1 — a × 1 = 1 × a = a    |   a × 0 = 0
• Distributive law (over addition): a × (b + c) = a × b + a × c   e.g. 2 × (4 + 7) = 2 × 4 + 2 × 7 = 22
• Distributive law (over subtraction): a × (b − c) = a × b − a × c   e.g. 2 × (7 − 4) = 2 × 7 − 2 × 4 = 6
• Division: a whole number divided by 1 is the number itself; 0 divided by any non-zero whole number is 0; division by 0 is not defined.

2. Exercise 2.1 — Solved

Q1. Write, if possible:

(i) The smallest natural number

Ans: 1

(ii) The smallest whole number

Ans: 0

(iii) The largest natural number

Ans: Not possible (the set is infinite)

(iv) The largest whole number

Ans: Not possible (the set is infinite)

Q2. Write first ten natural numbers.

Ans: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Q3. Write first ten whole numbers.

Ans: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Q4. Represent on number line:

(i) Whole numbers > 8

Ans: 9, 10, 11, 12, ... (points to the right of 8)

(ii) Whole numbers < 8

Ans: 0, 1, 2, 3, 4, 5, 6, 7

(iii) Whole numbers ≥ 8

Ans: 8, 9, 10, 11, ...

(iv) Whole numbers ≤ 8

Ans: 0, 1, 2, 3, 4, 5, 6, 7, 8

(v) Whole numbers > 3 but < 15

Ans: 4, 5, 6, ..., 14

(vi) Whole numbers > 5 but ≤ 12

Ans: 6, 7, 8, 9, 10, 11, 12

(vii) Whole numbers ≥ 4 but ≤ 11

Ans: 4, 5, 6, 7, 8, 9, 10, 11

(viii) Whole numbers ≥ 3 but < 9

Ans: 3, 4, 5, 6, 7, 8

(ix) Odd whole numbers greater than 3

Ans: 5, 7, 9, 11, ...

(x) Even whole numbers greater than or equal to 8 but less than 16

Ans: 8, 10, 12, 14

Q5. Find the sum of the following whole numbers by using number line:

(i) 1 and 5	→ 1 + 5 = 6
(ii) 6 and 3	→ 6 + 3 = 9
(iii) 10 and 2	→ 10 + 2 = 12
(iv) 8 and 4	→ 8 + 4 = 12
(v) 2, 3 and 5	→ 2 + 3 + 5 = 10
(vi) 3, 2 and 4	→ 3 + 2 + 4 = 9

Activity 1 (p. 125) — Fill in the blanks using Commutative and Associative laws of addition

(i) 56123 + 71045 = 71045 + 56123  (Commutative)

(ii) 24125 + (41625 + 7123) = (24125 + 41625) + 7123  (Associative)

(iii) 47813 + 51623 = 51623 + 47813  (Commutative)

(iv) 567 + (962 + 1784) = (567 + 962) + 1784  (Associative)

3. Exercise 2.2 — Solved

Q1. What is the sum of the largest number of six digits and the smallest number of seven digits?

Ans: Largest 6-digit = 999999, Smallest 7-digit = 1000000
Sum = 999999 + 1000000 = 1,999,999

Q2. Population of a village is 2700. If 1070 are men and 915 are women, find the number of children.

Ans: Children = 2700 − (1070 + 915) = 2700 − 1985 = 715 children

Q3. Arif deposited Rs 45800 in his bank account. After a month he withdrew Rs 3500 from it. How much money was left in his account?

Ans: Money left = 45800 − 3500 = Rs 42,300

Q4. Kashif had Rs 82000. He gave Rs 6500 to his wife, Rs 10550 to son and Rs 15335 to daughter. How much money was left with him?

Ans: Left = 82000 − (6500 + 10550 + 15335) = 82000 − 32385 = Rs 49,615

Q5. Verify commutative law under addition for the following:

(i) 7628 and 39780	7628 + 39780 = 47408 = 39780 + 7628 ✓
(ii) 924981 and 228	924981 + 228 = 925209 = 228 + 924981 ✓
(iii) 29000 and 10699	29000 + 10699 = 39699 = 10699 + 29000 ✓
(iv) 50102 and 9019854	50102 + 9019854 = 9069956 = 9019854 + 50102 ✓

Q6. Verify associative law under addition for the following:

(i) 34006, 2389 and 44380	(34006 + 2389) + 44380 = 80775 = 34006 + (2389 + 44380) ✓
(ii) 583031, 127 and 3405	(583031 + 127) + 3405 = 586563 = 583031 + (127 + 3405) ✓
(iii) 231, 6090 and 25996	(231 + 6090) + 25996 = 32317 = 231 + (6090 + 25996) ✓
(iv) 412, 3007 and 102341	(412 + 3007) + 102341 = 105760 = 412 + (3007 + 102341) ✓

Q7. Fill in the blanks to make the following statements true:

(i) 5020 + 849 =     + 5020

Ans: 849

(ii) 97864 + 0 =    

Ans: 97864

(iii) 749 + 0 = 0 +     =    

Ans: 749, 749

(iv) 3971 + (430 + 300) = (3971 +    ) + 300

Ans: 430

(v) 4853 + 93 = 93 + 4853 =    

Ans: 4946

Q8. Think a number. Double the number and add 9 to it, then add the number you started with. Then divide the amount by 3 and also subtract 3 from the quotient. What number do you get?

Ans: You get the starting number.
Why: number → 2n → 2n + 9 → + n = 3n + 9 → ÷ 3 = n + 3 → − 3 = n

4. Exercise 2.3 — Solved

Q1. Find the following products of whole numbers:

(i) 854 × 96	= 81984
(ii) 736 × 103	= 75808
(iii) 256 × 1008	= 258048
(iv) 995 × 158	= 157210

Q2. Divide and find the quotient and remainder:

(i) 7772 ÷ 58	Quotient = 134, Remainder = 0
(ii) 96324 ÷ 245	Quotient = 393, Remainder = 39
(iii) 16025 ÷ 1000	Quotient = 16, Remainder = 25
(iv) 92845 ÷ 300	Quotient = 309, Remainder = 145

Q3. A gardener plans to plant 570 trees in 19 rows. Each row should contain equal number of trees. How many trees will be in each row?

Ans: 570 ÷ 19 = 30 trees in each row

Q4. The shopkeeper purchased 125 television sets. If the cost of each set is Rs 9820, find the cost of all sets.

Ans: 125 × 9820 = Rs 1,227,500

Q5. Find the greatest 5 digit number which is exactly divisible by 75.

Ans: Greatest 5-digit number = 99999;  99999 ÷ 75 = 1333 remainder 24
99999 − 24 = 99975 (99975 ÷ 75 = 1333)

Q6. Multiply the greatest number of four digits with the smallest number of three digits.

Ans: 9999 × 100 = 999900

5. Exercise 2.4 — Solved

Q1. Fill in the blanks:

(i) 5 × 0 =    

Ans: 0

(ii) 6 × 1 =    

Ans: 6

(iii) 2 × 5 = 5 ×    

Ans: 2

(iv) 3 × (1 × 4) = (3 × 1) ×    

Ans: 4

(v) 5 × (     × 8) = (     × 7) × 8

Ans: 7, 5

(vi) 541 × (645 +    ) = 541 × 645 + 541 × 964

Ans: 964

(vii) 345 × (650 − 125) = 345 × 650     345 ×    

Ans: − (subtraction), 125

Q2. Write True or False:

(i) 545 × 248 = 545 + 248  — False
(ii) 67 − (125 − 12) = (67 − 125) − 12  — False
(iii) 64 × (245 − 10) = 64 × 245 − 64 × 10  — True (distributive law)
(iv) 95 × (25 × 14) = (95 × 25) + 14  — False

Q3. Verify commutative law under addition for 530 and 750.

Ans: LHS = 530 + 750 = 1280  and  RHS = 750 + 530 = 1280
Since LHS = RHS, commutative law under addition is verified.

Q4. Name and verify the law used in each of the following:

(i) 9 × 7 = 7 × 9	Law: Commutative law under multiplication  (63 = 63)
(ii) 4 × (6 × 3) = (4 × 6) × 3	Law: Associative law under multiplication  (72 = 72)
(iii) 5 × (7 − 1) = 5 × 7 − 5 × 1	Law: Distributive law of multiplication over subtraction  (30 = 30)
(iv) 2 × (8 + 3) = 2 × 8 + 2 × 3	Law: Distributive law of multiplication over addition  (22 = 22)

Q5. Verify Associative Law under multiplication for 240, 425 and 35.

Ans: LHS = (240 × 425) × 35 = 102000 × 35 = 3,570,000
RHS = 240 × (425 × 35) = 240 × 14875 = 3,570,000
Since LHS = RHS, associative law under multiplication is verified.

Q6. Verify distributive law of multiplication over addition for 300, 615 and 975.

Ans: LHS = 300 × (615 + 975) = 300 × 1590 = 477,000
RHS = 300 × 615 + 300 × 975 = 184,500 + 292,500 = 477,000
Since LHS = RHS, distributive law is verified.

6. Review Exercise 2 — Solved

Q1. Fill in the blanks by using laws under multiplication of whole numbers:

(i) 2 ×    = 3 ×     →  3, 2  (Commutative)

(ii) 112 × 528 =    ×     →  528, 112  (Commutative)

(iii) 5 × (3 × 8) = (5 × 3) ×     →  8  (Associative)

(iv)    × 7 = 7 ×    = 0  →  0, 0

(v) 9 × (2 + 16) =    × 2 +    × 16  →  9, 9  (Distributive)

(vi) (20 + 4) × 1 =    × 1 + 4 ×     →  20, 1  (Distributive)

(vii) 17 × (8 − 3) = 17 ×    −    × 3  →  8, 17  (Distributive)

(viii) (12 − 6) × 32 = 12 ×    − 6 ×     →  32, 32  (Distributive)

Q2. Fill in the blanks:

(i) 5 × 4 =    × 5  →  4

(ii) 15 × (10 × 6) = (    × 10) × 6  →  15

(iii) 10 × (15 + 6) = (10 × 15) + (    × 6)  →  10

(iv) 10 × (100 + 60) = (    × 100) + (10 ×   )  →  10, 60

Q3. Write True or False:

(i) 5 − 3 = 3 − 5  — False (2 ≠ −2)
(ii) 3 × 1 + 7 = 3 × 8  — False (10 ≠ 24)
(iii) 2 − (0 − 8) = (2 − 0) + 8  — True (2 + 8 = 2 + 8)
(iv) 9 + (7 + 5) = (9 + 7) + 5  — True (associative law)
(v) 4 × (35 × 2) = (4 × 35) × 2  — True (associative law)
(vi) 24 − (50 − 6) = (24 − 50) − 6  — False (−20 ≠ −32)
(vii) 14 ÷ 0 = 14  — False (division by 0 is not defined)
(viii) 0 ÷ 125 = 0  — True
(ix) 18 ÷ 18 = 0  — False (18 ÷ 18 = 1)
(x) 75 ÷ 75 = 1  — True

Q4. Write the predecessor and successor of the following numbers:

(i) 671

Ans: Predecessor = 670, Successor = 672

(ii) 245

Ans: Predecessor = 244, Successor = 246

Q5. Choose the correct answer:

(i) The smallest natural number is ________.  (a) 0  (b) 1  (c) 2  (d) 100

Answer: (b) 1

(ii) The predecessor of 1 in the set of whole numbers is ________.  (a) 0  (b) 2  (c) 3  (d) none

Answer: (a) 0

(iii) The smallest seven digit number is ________.  (a) 1234567  (b) 9999999  (c) 1111111  (d) 1000000

Answer: (d) 1000000

(iv) The greatest six digit number is ________.  (a) 876543  (b) 999999  (c) 111111  (d) 100000

Answer: (b) 999999

(v) The numbers divisible by 2 are called ________ numbers.  (a) prime  (b) even  (c) odd  (d) whole

Answer: (b) even

Q6. Draw a number line to represent the following whole numbers:

(i) 0, 1, 3, 9

Ans: Mark the points 0, 1, 3 and 9 on the number line (dark shaded dots).

(ii) Whole numbers > 3

Ans: 4, 5, 6, ... (measure to the right of 3)

(iii) Whole numbers ≤ 8

Ans: 0, 1, 2, 3, 4, 5, 6, 7, 8

(iv) Whole numbers > 5 but < 10

Ans: 6, 7, 8, 9

(v) Whole numbers ≥ 1 but ≤ 8

Ans: 1, 2, 3, 4, 5, 6, 7, 8

Q7. Find the sum of 4, 2, 3 and 5 on number line.

Ans: 4 + 2 + 3 + 5 = 14

Q8. Find two whole numbers whose sum is:

(i) 9 → e.g. 4 and 5
(ii) 24 → e.g. 10 and 14
(iii) 31 → e.g. 15 and 16
Note: any pair of whole numbers adding to the given sum is correct.

Q9. Verify commutative laws of addition and multiplication for 190 and 330.

Ans: Addition: 190 + 330 = 520 = 330 + 190 ✓
Multiplication: 190 × 330 = 62700 = 330 × 190 ✓

Q10. Verify Associative laws of addition and multiplication for 20, 30 and 60.

Ans: Addition: (20 + 30) + 60 = 110 = 20 + (30 + 60) ✓
Multiplication: (20 × 30) × 60 = 36000 = 20 × (30 × 60) ✓

Q11. Verify distributive laws of multiplication over addition and subtraction for 700, 500 and 100.

Ans: Over addition: 700 × (500 + 100) = 700 × 600 = 420,000
           700 × 500 + 700 × 100 = 350,000 + 70,000 = 420,000 ✓
Over subtraction: 700 × (500 − 100) = 700 × 400 = 280,000
           700 × 500 − 700 × 100 = 350,000 − 70,000 = 280,000 ✓

Q12. What is the cost of 640 photocopies at Rs 1.50 per copy?

Ans: 640 × 1.50 = Rs 960

Q13. Find the greatest number of 4 digits which is exactly divisible by 44.

Ans: Greatest 4-digit number = 9999;  9999 ÷ 44 = 227 remainder 11
9999 − 11 = 9988 (9988 ÷ 44 = 227, remainder 0)

7. Key Facts from the Summary (Book p. 135)

• Natural numbers N = {1, 2, 3, ...} are used for counting.
• Whole numbers W = {0, 1, 2, ...} = zero + all natural numbers.
• Sum and product of two whole numbers is also a whole number.
• Zero (0) is the additive identity; One (1) is the multiplicative identity.
• Addition and multiplication of whole numbers are commutative and associative.
• Multiplication is distributive over addition and subtraction (with positive difference).
• Division of two whole numbers gives a whole number only when the divisor divides exactly (remainder = 0).

Questions and answers verified against the ANSWER KEY of the Sindh Textbook Board (STBB) – Mathematics for Class 6, Unit 2: Whole Numbers (answer key on printed pp. 248–250 of the textbook).
Note: The workbook/guide (.doc) supplied for this unit is a scanned image file with no text layer, so it could not be extracted; all solutions above are worked out step-by-step and then cross-checked against the printed answer key. All checked answers agree with the key (e.g. products 81984 / 75808 / 258048 / 157210; quotients 134r0, 393r39, 16r25, 309r145; Rs 42,300; 715 children; 99975; 999900; Rs 1,227,500; 9988; Rs 960).
Answers that depend on choice (Review Q8 pairs) are given as examples only. Prepared for study & revision purpose only.