Commutative Property

Order Does Not Matter

The Commutative Property states that changing the order of the terms in an addition or multiplication expression does not alter the final result. The terms can swap places freely.

Key Points

The Commutative Property of Addition is written as $a + b = b + a$ .
The Commutative Property of Multiplication is written as $a \times b = b \times a$ .
Example: $2 + 4 = 4 + 2 = 6$ and $3 \times 4 = 4 \times 3 = 12$ .
This property does not apply to subtraction or division.

Olympiad Fact

In competitive math, we use the commutative property to reorder messy sequences so that compatible numbers (like $17$ and $83$ ) sit next to each other for easy mental addition.

Associative Property

Grouping Does Not Matter

The Associative Property states that when adding or multiplying three or more numbers, the way they are grouped inside parentheses does not affect the final result.

Key Points

The Associative Property of Addition is $(a + b) + c = a + (b + c)$ .
The Associative Property of Multiplication is $(a \times b) \times c = a \times (b \times c)$ .
Grouping does not change the final mathematical sum or product.
This property lets you group numbers in the most convenient way to calculate faster.

Olympiad Fact

When adding a long list of numbers, use the associative property to group pairs that add up to multiples of $10$ (like grouping $4$ and $6$ first).

The Distributive Property

Multiplication Over Sums

The Distributive Property explains how multiplication interacts with addition and subtraction inside parentheses. Multiplying a number by a sum (or difference) is the same as multiplying it by each term individually.

Key Points

Distribution over addition: $a \times (b + c) = a \times b + a \times c$ .
Distribution over subtraction: $a \times (b - c) = a \times b - a \times c$ .
Example: $5 \times (12 + 8)$ is equal to $5 \times 12 + 5 \times 8$ .
This property is essential for breaking down complex calculations.

Olympiad Fact

The distributive property is the mathematical bridge that allows us to factor out common terms in algebraic equations.

Reverse Distributivity (Factoring Out)

Grouping Common Multipliers

Factoring is the reverse of the distributive property. When different terms share an identical multiplier, we can pull that factor outside the parentheses to simplify the calculation.

Key Points

Factoring out is the exact reverse of the distributive property.
Formula: $a \times b + a \times c = a \times (b + c)$ .
Example: $15 \times 4 + 15 \times 6$ is simplified by pulling out the common factor $15$ .
This transforms the expression into $15 \times (4 + 6) = 15 \times 10 = 150$ .

Olympiad Fact

Olympiad problems often present massive, scary-looking expressions; factoring out common terms is the secret to solving them in seconds.

Fast Calculations

Mental Math Shortcuts

We can use the distributive property to perform difficult multiplication in our heads by rewriting numbers close to

10

,

50

, or

100

as a sum or difference.

Key Points

Use the distributive property to perform complex mental math.
Rewrite difficult multipliers close to a base-10 number.
Example: To multiply $25 \times 97$ , rewrite $97$ as $(100 - 3)$ .
Distribute: $25 \times (100 - 3) = 25 \times 100 - 25 \times 3 = 2500 - 75 = 2425$ .

Olympiad Fact

Using distribution to round numbers to the nearest base-10 makes solving large multiplication questions fast, accurate, and completely mental.