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.
We must follow a strict, standard order to find the single correct value.

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

$\mathbf{B}$ stands for Brackets, which refers to parentheses $(\quad)$ . Brackets are evaluated from the innermost set to the outermost set if nested.
$\mathbf{O}$ stands for 'Of', representing a direct form of multiplication.
$\mathbf{D}$ and $\mathbf{M}$ stand for Division and Multiplication, calculated left-to-right.
$\mathbf{A}$ and $\mathbf{S}$ stand for Addition and Subtraction, calculated left-to-right.
This structured sequence from $\mathbf{B}$ to $\mathbf{S}$ guarantees a unique, correct mathematical result.

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 fundamental grouping symbols in arithmetic.
They tell you to solve the enclosed numbers first.
Example: In $5 \times (12 + 8)$ , add $12 + 8$ first to get $20$ .
Then multiply: $5 \times 20 = 100$ .
Without parentheses, $5 \times 12 + 8$ would incorrectly equal $60 + 8 = 68$ .

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'.
This translates to $(5 + 7) \times 6$ , which equals $72$ .
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.

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