Practice Maths

L41 — Number Laws

Key Terms

commutative law
Order doesn’t matter for addition and multiplication: a + b = b + a and a × b = b × a. Does NOT apply to subtraction or division.
associative law
Grouping doesn’t matter for addition and multiplication: (a + b) + c = a + (b + c). Does NOT apply to subtraction or division.
distributive law
Multiply each term inside the bracket: a(b + c) = ab + ac. Used for mental multiplication and expanding algebraic expressions.
identity law
Adding 0 leaves a number unchanged (additive identity). Multiplying by 1 leaves a number unchanged (multiplicative identity): a + 0 = a and a × 1 = a.

The Four Number Laws

Commutative Law: order doesn’t matter for + and ×.

  • a + b = b + a  (e.g. 3 + 7 = 7 + 3)
  • a × b = b × a  (e.g. 4 × 6 = 6 × 4)
  • Note: subtraction and division are NOT commutative.

Associative Law: grouping doesn’t matter for + and ×.

  • (a + b) + c = a + (b + c)
  • (a × b) × c = a × (b × c)

Distributive Law: multiply each term inside the bracket.

  • a(b + c) = ab + ac

Identity Law: 0 is the additive identity; 1 is the multiplicative identity.

  • a + 0 = a    a × 1 = a
Hot Tip: The distributive law is the key to mental arithmetic. 6 × 23 = 6 × 20 + 6 × 3 = 120 + 18 = 138. Break hard numbers into easy ones!

Worked Examples

Example 1: Use the distributive law to calculate 7 × 18.

7 × 18 = 7 × (20 − 2) = 7 × 20 − 7 × 2 = 140 − 14 = 126

Example 2: Simplify 4 × 25 + 4 × 75.

= 4 × (25 + 75) = 4 × 100 = 400

Example 3: Calculate 9 × 99 mentally.

9 × 99 = 9 × (100 − 1) = 900 − 9 = 891

Why Do We Need Number Laws?

Number laws describe rules that are always true for any numbers. They’re not arbitrary — they arise naturally from the way numbers behave. Understanding them explains why mental arithmetic shortcuts work, and they form the foundation for everything in algebra.

The Commutative Law: Order Doesn’t Matter

For addition and multiplication, you can swap the order and get the same answer: 3 + 7 = 7 + 3, and 4 × 6 = 6 × 4. It’s why you can rearrange numbers to make mental maths easier.

But beware: subtraction and division are NOT commutative. 8 − 3 = 5, but 3 − 8 = −5. And 12 ÷ 4 = 3, but 4 ÷ 12 = 13. Always check which operation you’re using before swapping numbers.

The Associative Law: Grouping Doesn’t Matter

For addition and multiplication, you can change how you group numbers without changing the result: (2 + 3) + 4 = 2 + (3 + 4). Both equal 9.

Practical use: 25 + 37 + 75. Notice 25 + 75 = 100. Regroup: (25 + 75) + 37 = 100 + 37 = 137. Mental calculation made easy!

The Distributive Law: The Mental Maths Superpower

This is the most practically useful law: a × (b + c) = a × b + a × c. Break a hard multiplication into two easy ones:

  • 7 × 18 = 7 × (20 − 2) = 140 − 14 = 126
  • 6 × 45 = 6 × (40 + 5) = 240 + 30 = 270

Every time you do mental multiplication by splitting a number into tens, you’re using the distributive law.

Remember: The number outside the brackets multiplies every term inside. a(b + c) = ab + ac. This is the foundation of expanding brackets in algebra.

The Identity Law: Adding Nothing, Multiplying by One

Adding 0 leaves a number unchanged: a + 0 = a. Multiplying by 1 leaves a number unchanged: a × 1 = a. These “identity elements” are crucial in algebra when solving equations.

Common Mistake: Misapplying the commutative law to subtraction. “5 − 3 = 3 − 5” is FALSE. Order matters for subtraction. The commutative law ONLY applies to addition and multiplication.

  1. Apply the Distributive Law (Mental Arithmetic)

    For (a)–(c), the split has been given to get you started. For (d)–(h), choose your own split and show your working the same way.

    1. Solve 6 × 23 by rewriting it as 6 × (20 + 3), then applying the distributive law.
    2. Solve 7 × 18 by rewriting it as 7 × (10 + 8), then applying the distributive law.
    3. Solve 9 × 99 by rewriting it as 9 × (100 − 1), then applying the distributive law.
    4. 4 × 52
    5. 8 × 31
    6. 5 × 47
    7. 3 × 98
    8. 6 × 45

  2. Apply Commutative and Associative Laws

    In each expression, two of the three numbers pair up to make a round number (like 10, 20, 100, or 200). Use the commutative law to swap them next to each other, calculate that pair first, then find the final answer.

    Example:   13 + 58 + 7  →  spot that 13 + 7 = 20  →  (13 + 7) + 58 = 20 + 58 = 78
    1. 17 + 38 + 3
    2. 25 × 7 × 4
    3. 46 + 19 + 54
    4. 5 × 13 × 2
    5. 34 + 27 + 66
    6. 4 × 9 × 25
    7. 125 + 87 + 75
    8. 2 × 37 × 5

  3. Expand Using the Distributive Law

    Expand each expression by multiplying the number outside the bracket by every term inside.

    Example (numbers):   5(3 + 4) → 5 × 3 + 5 × 4 = 15 + 20 = 35
    Example (algebra):   3(x + 2) → 3 × x + 3 × 2 = 3x + 6
    1. 3(4 + 5)
    2. 5(2 + 7)
    3. 4(10 − 3)
    4. 6(5 + 8)
    5. 2(x + 4)
    6. 5(2x + 3)
    7. 3(4y − 2)
    8. 7(a + b)

  4. Identify Laws and True/False Statements

    Recall the three laws:
    Commutative: order doesn’t matter — a + b = b + a   and   a × b = b × a
    Associative: grouping doesn’t matter — (a + b) + c = a + (b + c)
    Distributive: multiply through the bracket — a(b + c) = ab + ac
    1. Which law justifies: 3 × 7 = 7 × 3?
    2. Which law justifies: (2 + 5) + 8 = 2 + (5 + 8)?
    3. Which law justifies: 4(3 + 6) = 4 × 3 + 4 × 6?
    4. True or False: 10 − 6 = 6 − 10. Explain.
    5. True or False: 12 ÷ 4 = 4 ÷ 12. Explain.
    6. Give a counterexample to show subtraction is not commutative.
    7. True or False: a × 0 = 0 for any number a.
    8. True or False: a + 0 = a for any number a.

  5. Problem Solving with Number Laws

    Example:   4 bags each containing 6 apples and 3 bananas → 4 × (6 + 3) = 4 × 6 + 4 × 3 = 24 + 12 = 36 items (distributive law)
    1. A shop sells 8 boxes each containing 15 red pens and 5 blue pens. Use the distributive law to find the total number of pens.
    2. Sam calculates 4 × 25 × 13. What is the easiest order to multiply these, and why? What is the answer?
    3. Show two different ways to calculate 6 × 37 using the distributive law.
    4. Explain why the associative law does not work for division using the example (24 ÷ 6) ÷ 2 versus 24 ÷ (6 ÷ 2).
    5. Use the distributive law to write a simplified expression for: n(n + 3) + n(n − 1).

  6. Factorise Using the Distributive Law

    Write each expression as a product of a common factor and a bracket (reverse of expanding).

    Example:   8x + 12 → HCF of 8 and 12 is 4 → 4(2x + 3)
    Check: 4 × 2x + 4 × 3 = 8x + 12  ✓
    1. 6x + 9
    2. 10a + 15
    3. 12y − 8
    4. 20m + 5
    5. 3p + 3q
    6. 4x + 6y + 10
    7. 14 − 21n
    8. 9a + 12b + 15c

  7. Apply Laws to Simplify Algebraic Expressions

    Simplify each expression by collecting like terms or applying a number law. State which law you used.

    Example (like terms):   3x + 5x → (3 + 5)x = 8x
    Example (associative law):   4 × (n × 5) → (4 × 5) × n = 20n
    1. 3x + 4x
    2. 5y × 1
    3. 7a + 0
    4. 4 × (n × 5)
    5. 2x + 3y + 5x
    6. (4a + 3a) + 2a
    7. 6 × m × 0
    8. 3(2n) + 4n

  8. Order of Operations with Number Laws

    Evaluate each expression by applying BODMAS and number laws.

    BODMAS order:   Brackets → Orders (powers) → Division → Multiplication → Addition → Subtraction
    Example:   4 + 2 × 32 → 4 + 2 × 9 → 4 + 18 = 22
    1. 3 + 4 × 5
    2. (3 + 4) × 5
    3. 2 × 32 + 4
    4. (2 × 3)2 + 4
    5. 24 ÷ (4 + 2) − 1
    6. 5 + 3 × (8 − 5)
    7. 42 − (2 + 3) × 2
    8. 7 × 8 + 7 × 2   (use distributive law to simplify first)

  9. Number Law Investigation

    1. A student claims: “(a − b) − c = a − (b − c) always.” Is this true? Test with a = 20, b = 8, c = 3, and explain why the associative law does not apply to subtraction.
    2. Use the distributive law to show that 99 × 37 = 100 × 37 − 37. Calculate the answer mentally.
    3. Explain why 5 × (3 × 4) and (5 × 3) × 4 give the same result. Which law does this demonstrate?
    4. Expand and simplify: 4(x + 3) + 2(x + 5).
    5. A rectangle has length (3x + 5) and width 4. Write an expression for its perimeter and one for its area. Simplify both.

  10. Mental Arithmetic Strategies

    Use number laws to calculate each problem mentally. Show the strategy used.

    1. 29 × 8   (hint: (30 − 1) × 8)
    2. 4 × 17 × 25   (hint: rearrange)
    3. 198 + 345 + 2   (hint: rearrange)
    4. 6 × 36 + 6 × 14
    5. 50 × 13 × 2
    6. 7 × 104   (hint: (100 + 4) × 7)
    7. 125 × 8   (hint: 125 = 1000 ÷ 8)
    8. Find the missing number: ☐ × (15 + 25) = 240
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