LESSON 11
ORDER OF OPERATIONS
The Rules That Keep Math Consistent
Why Does Order Matter?
When a math expression contains more than one operation, the order in which you perform them matters enormously. Without agreed-upon rules, the same expression could produce different answers depending on who solves it.
The order of operations is a universal set of rules that tells us exactly which operations to perform first. In the United States these rules are remembered using the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
PEMDAS Explained
P — Parentheses: always solve what is inside grouping symbols first. E — Exponents: evaluate all powers and roots next. M/D — Multiplication and Division: perform these left to right as they appear. A/S — Addition and Subtraction: perform these left to right as they appear last. Multiplication does not automatically come before division — they share equal priority and are worked left to right.
Working Through an Expression
To solve a complex expression, identify each operation and apply PEMDAS step by step. Work through one level at a time, rewriting the expression after each step. This systematic approach prevents errors and makes your work easy to check.
Nested Parentheses
When parentheses appear inside other parentheses, work from the innermost set outward. Brackets [ ] and braces { } are also used as grouping symbols and follow the same rule — always resolve the innermost group first.
Common Mistakes to Avoid
The most common errors are: performing addition before multiplication, treating multiplication as always coming before division (they are equal priority), and forgetting to work left to right for same-priority operations. Always rewrite the expression after each step to stay organized.
Key Facts
PEMDAS is also remembered as 'Please Excuse My Dear Aunt Sally'. In the UK and other countries the same rules are called BODMAS or BIDMAS.
Multiplication and division have equal priority — so do addition and subtraction. When two equal-priority operations appear, always work left to right.
A fraction bar acts as a grouping symbol. Everything above the bar is one group and everything below is another — evaluate each side before dividing.
Calculators follow the order of operations automatically, but only if you enter the expression correctly. Misplaced parentheses are a leading cause of calculator errors.