Order of Operations and Parentheses

Complex Expressions

Multiple Operations, One Puzzle

When an arithmetic expression contains more than one basic mathematical operation (such as combining addition, subtraction, multiplication, and division), it is called a complex expression.

Key Points

Complex expressions combine multiple operations in one statement.
They contain mixtures of $+$ , $-$ , $\times$ , or $\div$ .
Solving them without rules leads to wrong, conflicting answers.
Example: $6 + 3 \times 4$ is not solved by adding $6 + 3$ first.

Olympiad Fact

A complex expression without operations rules is mathematically ambiguous; order-of-operation laws are the grammar of arithmetic.

The BODMAS Rule

The Order of Operations

The BODMAS rule defines the strict mathematical hierarchy used to solve complex expressions. It ensures that everyone evaluates an expression to the exact same unique value.

Key Points

BODMAS dictates the exact, step-by-step order for evaluating operations.
First, evaluate parentheses $(\quad)$ , solving from the innermost set to the outermost set if nested.
Second, resolve 'Of' operations (which behave as direct multiplication) after parentheses are cleared.
Third, execute division ( $\div$ ) and multiplication ( $\times$ ) in order from left to right, then addition and subtraction.

Olympiad Fact

Division and Multiplication hold equal priority, as do Addition and Subtraction. If they appear next to each other, you must solve them in order from left to right!

The Role of Parentheses

Prioritizing Operations

Parentheses

(\quad)

are grouping symbols. They tell us exactly which operations to perform first, overriding the standard order of operations when necessary.

Key Points

Parentheses $(\quad)$ are the primary grouping symbols used to override standard priorities.
If parentheses are nested, always evaluate the innermost set of parentheses first.
Example: In $5 \times (12 + 8)$ , the addition must be solved first to obtain $20$ .
After resolving the parentheses, complete the outer operation: $5 \times 20 = 100$ .

Olympiad Fact

In complex Olympiad problems, brackets are strategically placed to group terms that cancel each other out, helping you avoid long, messy calculations.

Evaluating Multiplicative Terms First

Solving Terms Before Sums

If a complex expression contains multiplication or division but has no parentheses, we evaluate each multiplication or division term first before performing any addition or subtraction.

Key Points

Multiplicative operations bind numbers together tightly as single terms.
Never perform addition before adjacent multiplication unless parentheses dictate it.
Example: In $20 + 2 \times 6$ , do not add $20 + 2$ first.
First solve the term $2 \times 6 = 12$ , then finish the addition: $20 + 12 = 32$ .

Olympiad Fact

This rule explains why calculators and computers are programmed to automatically execute multiplication and division before addition and subtraction.

Translating Words to Expressions

Modeling Real-World Phrases

Parentheses are crucial when translating verbal descriptions into mathematical expressions. They ensure that operations described as happening first are calculated first.

Key Points

Words tell us the required order of operations.
Phrases like 'sum of' or 'difference of' often need parentheses.
Example: 'Multiply the sum of 5 and 7 by 6' translates to $(5 + 7) \times 6$ .
Without parentheses, $5 + 7 \times 6$ incorrectly does multiplication first.

Olympiad Fact

'5 more than the product of 7 and 6' translates to $5 + 7 \times 6$ . No parentheses are needed here because multiplication naturally happens first under BODMAS.

Swipe to skip‹