Order of Operations — PEMDAS / BODMAS Rules

Hand the expression $24 / 3 \times (1 + 3) - 2^2$ to two people with no shared rule and you can get two different answers. Order of operations is the agreement that removes that ambiguity, and PEMDAS or BODMAS is the name for the agreement.

The rule and its symbols

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. BODMAS swaps in Brackets and Orders for the first two, but it is the same rule. Written as stages:

1. Simplify inside grouping symbols such as parentheses (innermost first if nested).
2. Evaluate exponents.
3. Do multiplication and division from left to right.
4. Do addition and subtraction from left to right.

A fraction bar also acts like grouping, because the whole numerator and denominator stay together.

Why it is four stages, not six

The acronym lists six letters, but multiplication and division are actually one stage, and so are addition and subtraction. The reason is mathematical, not arbitrary: division is multiplication by a reciprocal, and subtraction is addition of a negative. Since $a \div b = a \times \tfrac{1}{b}$, there is no real "divide before multiply" choice to make; you just clear that one stage left to right. The same logic ties addition and subtraction together.

That is why PEMDAS does not say "always multiply before dividing." It says "finish the multiply-or-divide stage in order, left to right."

Worked example: one stage at a time

Evaluate

Start with the parentheses:

Evaluate the exponent:

Now division and multiplication, left to right. Since $24/3 = 8$:

Multiply, then subtract:

So $24 / 3 \times (1 + 3) - 2^2 = 28$. The trap is visible here: multiplying $3 \times 4$ first would break the left-to-right rule inside the multiply-and-divide stage.

Practice and self-check

Work this one stage at a time:

Parentheses give $30 / 5 \times 3 + 3^2$; the exponent gives $30 / 5 \times 3 + 9$; left-to-right division and multiplication give $6 \times 3 + 9 = 18 + 9$; addition gives $\mathbf{27}$. If you got a different number, check whether you divided before you multiplied, since $30/5$ comes first from the left. Before moving on, ask two questions: did I clear grouping and exponents first, and did I move left to right inside each of the two basic stages? Two yeses means the structure is sound.

Common traps

Reading PEMDAS as a strict ladder. In $20 / 5 \times 2$ you divide first because it appears first, so the result is $8$, not $2$.

Forcing addition before subtraction. In $10 - 3 + 1$ you work left to right and get $8$.

Skipping the rewrite step. Rewriting the expression after each stage takes a few seconds but prevents dropped signs, missed exponents, and operations done too early.

You use this rule whenever an expression mixes operations: school arithmetic, algebra, science formulas, spreadsheets, and calculator input. Programming languages use operator precedence too, and the exact symbols can differ by tool, but the core idea is the same: some operations are grouped before others so expressions are read consistently.