Number Laws — Solutions

1. Apply the Distributive Law (Mental Arithmetic)

   1. 6 × (20 + 3) = 6 × 20 + 6 × 3 = 120 + 18 = 138 ▶ View Solution
   2. 7 × (10 + 8) = 7 × 10 + 7 × 8 = 70 + 56 = 126 ▶ View Solution
   3. 9 × (100 − 1) = 9 × 100 − 9 × 1 = 900 − 9 = 891 ▶ View Solution
   4. 4 × 52: 208 ▶ View Solution
   5. 8 × 31: 248 ▶ View Solution
   6. 5 × 47: 235 ▶ View Solution
   7. 3 × 98: 294 ▶ View Solution
   8. 6 × 45: 270 ▶ View Solution

2. Apply Commutative and Associative Laws

   1. (17 + 3) + 38 = 20 + 38 = 58 ▶ View Solution
   2. (25 × 4) × 7 = 100 × 7 = 700 ▶ View Solution
   3. (46 + 54) + 19 = 100 + 19 = 119 ▶ View Solution
   4. (5 × 2) × 13 = 10 × 13 = 130 ▶ View Solution
   5. (34 + 66) + 27 = 100 + 27 = 127 ▶ View Solution
   6. (4 × 25) × 9 = 100 × 9 = 900 ▶ View Solution
   7. (125 + 75) + 87 = 200 + 87 = 287 ▶ View Solution
   8. (2 × 5) × 37 = 10 × 37 = 370 ▶ View Solution

3. Expand Using the Distributive Law

   1. 3(4 + 5): 27 ▶ View Solution
   2. 5(2 + 7): 45 ▶ View Solution
   3. 4(10 − 3): 28 ▶ View Solution
   4. 6(5 + 8): 78 ▶ View Solution
   5. 2(x + 4): 2x + 8 ▶ View Solution
   6. 5(2x + 3): 10x + 15 ▶ View Solution
   7. 3(4y − 2): 12y − 6 ▶ View Solution
   8. 7(a + b): 7a + 7b ▶ View Solution

4. Identify Laws and True/False Statements

   1. 3 × 7 = 7 × 3: Commutative Law of Multiplication ▶ View Solution
   2. (2 + 5) + 8 = 2 + (5 + 8): Associative Law of Addition ▶ View Solution
   3. 4(3 + 6) = 4 × 3 + 4 × 6: Distributive Law ▶ View Solution
   4. 10 − 6 = 6 − 10?: False — subtraction is not commutative ▶ View Solution
   5. 12 ÷ 4 = 4 ÷ 12?: False — division is not commutative ▶ View Solution
   6. Counterexample that subtraction is not commutative: 9 − 4 = 5 but 4 − 9 = −5 ▶ View Solution
   7. a × 0 = 0 for any a: True ▶ View Solution
   8. a + 0 = a for any a: True ▶ View Solution

5. Problem Solving with Number Laws

   1. 8 shops × (15 + 5) pens: 160 pens ▶ View Solution
   2. 4 × 17 × 25: 1 700 ▶ View Solution
   3. 6 × 37 using two methods: 222 ▶ View Solution
   4. Is division associative? Show (24 ÷ 6) ÷ 2 vs 24 ÷ (6 ÷ 2): Not associative — gives 2 vs 8 ▶ View Solution
   5. n(n + 3) + n(n − 1): 2n² + 2n ▶ View Solution

6. Factorise Using the Distributive Law

   1. 6x + 9: 3(2x + 3) ▶ View Solution
   2. 10a + 15: 5(2a + 3) ▶ View Solution
   3. 12y − 8: 4(3y − 2) ▶ View Solution
   4. 20m + 5: 5(4m + 1) ▶ View Solution
   5. 3p + 3q: 3(p + q) ▶ View Solution
   6. 4x + 6y + 10: 2(2x + 3y + 5) ▶ View Solution
   7. 14 − 21n: 7(2 − 3n) ▶ View Solution
   8. 9a + 12b + 15c: 3(3a + 4b + 5c) ▶ View Solution

7. Apply Laws to Simplify Algebraic Expressions

   1. 3x + 4x: 7x ▶ View Solution
   2. 5y × 1: 5y ▶ View Solution
   3. 7a + 0: 7a ▶ View Solution
   4. 4 × (n × 5): 20n ▶ View Solution
   5. 2x + 3y + 5x: 7x + 3y ▶ View Solution
   6. (4a + 3a) + 2a: 9a ▶ View Solution
   7. 6 × m × 0: 0 ▶ View Solution
   8. 3(2n) + 4n: 10n ▶ View Solution

8. Order of Operations with Number Laws

   1. 3 + 4 × 5: 23 ▶ View Solution
   2. (3 + 4) × 5: 35 ▶ View Solution
   3. 2 × 3² + 4: 22 ▶ View Solution
   4. (2 × 3)² + 4: 40 ▶ View Solution
   5. 24 ÷ (4 + 2) − 1: 3 ▶ View Solution
   6. 5 + 3 × (8 − 5): 14 ▶ View Solution
   7. 4² − (2 + 3) × 2: 6 ▶ View Solution
   8. 7 × 8 + 7 × 2: 70 ▶ View Solution

9. Number Law Investigation

   1. Is subtraction associative? Test (20 − 8) − 3 vs 20 − (8 − 3): Not associative — gives 9 vs 15 ▶ View Solution
   2. 99 × 37: 3 663 ▶ View Solution
   3. 5 × (3 × 4), name the law: 60 — Associative Law of Multiplication ▶ View Solution
   4. 4(x + 3) + 2(x + 5): 6x + 22 ▶ View Solution
   5. Rectangle perimeter and area (width 4, length 3x + 5): Perimeter: 6x + 18; Area: 12x + 20 ▶ View Solution

10. Mental Arithmetic Strategies

    1. 29 × 8: 232 ▶ View Solution
    2. 4 × 17 × 25: 1 700 ▶ View Solution
    3. 198 + 345 + 2: 545 ▶ View Solution
    4. 6 × 36 + 6 × 14: 300 ▶ View Solution
    5. 50 × 13 × 2: 1 300 ▶ View Solution
    6. 7 × 104: 728 ▶ View Solution
    7. 125 × 8: 1 000 ▶ View Solution
    8. ___ × 40 = 240: 6 ▶ View Solution