3. Properties of Whole Numbers

Properties of Whole Numbers

Whole numbers are non-negative integers (0, 1, 2, 3, …). Understanding their properties is foundational for tackling more advanced arithmetic and algebra problems. These properties help simplify complex expressions and support logical problem-solving.

Basic Properties

1. Closure Property
   • Whole numbers are closed under addition and multiplication. This means adding or multiplying two whole numbers always yields another whole number.
   • Example:
     • 3 + 5 = 8 (Whole number)
     • 4 × 6 = 24 (Whole number)
2. Commutative Property
   • The order of addition or multiplication doesn’t change the result.
   • Addition: a + b = b + a
   • Multiplication: a × b = b × a
   • Example:
     • 5 + 3 = 3 + 5
     • 4 × 7 = 7 × 4
3. Associative Property
   • Grouping does not affect the outcome in addition or multiplication.
   • Addition: ( a + b ) + c = a + ( b + c )
   • Multiplication: ( a × b ) × c = a × ( b × c )
   • Example:
     • (2 + 3 ) + 4 = 2 + ( 3 + 4 )
     • (3 × 2 ) × 5 = 3 × ( 2 × 5 )
4. Distributive Property
   • Multiplication distributes over addition.
   • a × ( b + c ) = a × b + a × c
   • Example:
     • 3 × ( 4 + 5 ) = 3 × 4 + 3 × 5
5. Identity Property
   • 0 is called an identity for addition of whole numbers or additive identity for whole numbers. The identity property of addition states that adding zero to a number doesn’t change it: a + 0 = a
   • 1 is called an identity for multiplication of whole numbers or multiplicative identity for whole numbers. The identity property of multiplication states that multiplying a number by 1 doesn’t change it: a × 1 = a

---

Progressing to Advanced Concepts

Now, we’ll explore scenarios and logic-based problems that combine these properties in complex ways.

Advanced Questions

1. Question:
If x = 7 and y = 12 verify the associative and distributive properties by finding:( x + y ) + 5 and x + ( y + 5 )

and

x × ( y + 2 ) and ( x × y ) + ( x × 2 )

Solution:

• For the associative property:
  • Left Side: ( 7 + 12 ) + 5 = 19 + 5 = 24
  • Right Side: 7 + ( 12 + 5 ) = 7 + 17 = 24
  • Both are equal, so the associative property holds.
• For the distributive property:
  • Left Side: 7 × ( 12 + 2 ) = 7 × 14 = 98
  • Right Side: ( 7 × 12 ) + ( 7 × 2 ) = 84 + 14 = 98
  • Both are equal, so the distributive property holds.

2. Question:
A school has 4 classes, each with 25 students. Each student needs 3 notebooks. Using properties, find the total number of notebooks required.

Solution:

• Total notebooks: 4 × ( 25 × 3 )
• By associative property: ( 4 × 25 ) × 3 = 100 × 3 = 300
• Alternatively: 4 × 75 = 300
• Answer: 300 notebooks.

---

More Questions

1. Question:
You are given that x = 15, y = 8, and z = 5. Using distributive property, verify:x × ( y + z ) = x × y + x × z

Solution:

• Left Side: 15 × ( 8 + 5 ) = 15 × 13 = 195
• Right Side: ( 15 × 8) + ( 15 × 5 ) = 120 + 75 = 195
• Answer: Both sides are equal, so the distributive property holds.

2. Question:
Suppose a warehouse arranges products in m rows with n items per row. If the rows and items per row are increased by 3, show that the increase in products is represented by:

3(m + n + 3)

Solution:

• New arrangement: (m + 3) × (n + 3) = mn + 3m + 3n + 9
• Original: mn
• Increase in products: (mn + 3m + 3n + 9) − mn = 3m + 3n + 9 = 3(m + n + 3)
• Answer: The increase in products is 3(m + n + 3)

---

10 Practice Questions with Solutions

1. Question:
   Verify if 5 × (7 + 4) = (5 × 7) + (5 × 4)
   Solution:
   • Left Side: 5 × 11 = 55
   • Right Side: 35 + 20 =55
   • Answer: Both are equal.
2. Question:
   Simplify (8 + 5) + (6 + 7) using the associative property.
   Solution:
   • By grouping: (8 + 5) + (6 + 7 ) = 13 + 13 = 26
   • Answer: 26.
3. Question:
   Check if (4+9)+6=4+(9+6)(4 + 9) + 6 = 4 + (9 + 6)(4+9)+6=4+(9+6).Solution:
   • Left Side: 13+6=1913 + 6 = 1913+6=19
   • Right Side: 4+15=194 + 15 = 194+15=19
   • Answer: True.
4. Question:
   If a=3a = 3a=3, b=5b = 5b=5, c=2c = 2c=2, find a×(b+c)a \times (b + c)a×(b+c).Solution:
   • 3×(5+2)=3×7=213 \times (5 + 2) = 3 \times 7 = 213×(5+2)=3×7=21
   • Answer: 21.
5. Question:
   Simplify 7×(3+2)+57 \times (3 + 2) + 57×(3+2)+5.Solution:
   • Using distributive: 7×5+5=35+5=407 \times 5 + 5 = 35 + 5 = 407×5+5=35+5=40
   • Answer: 40.
6. Question:
   Prove (6+4)×3=(6×3)+(4×3)(6 + 4) \times 3 = (6 \times 3) + (4 \times 3)(6+4)×3=(6×3)+(4×3).Solution:
   • Left Side: 10×3=3010 \times 3 = 3010×3=30
   • Right Side: 18+12=3018 + 12 = 3018+12=30
   • Answer: True.
7. Question:
   Evaluate 9×(5+1)−(9×5)9 \times (5 + 1) – (9 \times 5)9×(5+1)−(9×5).Solution:
   • Using distributive: 9×6−45=54−45=99 \times 6 – 45 = 54 – 45 = 99×6−45=54−45=9
   • Answer: 9.
8. Question:
   If each of 5 classes has 12 students and each student needs 3 pencils, find the total pencils needed using associative property.Solution:
   • (5×12)×3=60×3=180(5 \times 12) \times 3 = 60 \times 3 = 180(5×12)×3=60×3=180
   • Answer: 180 pencils.
9. Question:
   Check if (7×3)+(7×4)=7×(3+4)(7 \times 3) + (7 \times 4) = 7 \times (3 + 4)(7×3)+(7×4)=7×(3+4).Solution:
   • Left Side: 21+28=4921 + 28 = 4921+28=49
   • Right Side: 7×7=497 \times 7 = 497×7=49
   • Answer: True.
10. Question:
    A hall has 6 rows with 8 chairs each. If 4 more rows are added, calculate the total chairs using distributive property.Solution:
    • 6×8+4×8=(6+4)×8=10×8=806 \times 8 + 4 \times 8 = (6 + 4) \times 8 = 10 \times 8 = 806×8+4×8=(6+4)×8=10×8=80
    • Answer: 80 chairs.

Practice makes perfect. We recommend you to practice topic-wise tests https://rankmadeeasy.com/courses-and-test-series . Our expert team has prepared these tests to help you become confident in each topic.

Our Social Media Presence –
https://youtube.com/@rankmadeeasy
https://www.facebook.com/groups/rankmadeeasy
https://www.facebook.com/RankMadeEasy
https://www.instagram.com/rankmadeeasy
https://x.com/rankmadeeasy
https://linkedin.com/in/rankmadeeasy