BODMAS / PEMDAS — Order of Operations Explained

BODMAS, also written PEMDAS, sets the order for evaluating an expression that mixes operations: brackets or parentheses first, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.

The trap is assuming multiplication always beats division, or addition always beats subtraction. It does not. Each pair shares the same priority, so you move left to right.

Why the order is structured this way

The two names describe one structure with slightly different vocabulary:

1. $B$ or $P$: Brackets or Parentheses
2. $O$ or $E$: Orders or Exponents
3. $D$ and $M$: Division and Multiplication
4. $A$ and $S$: Addition and Subtraction

They are not different systems, just two mnemonics for the same order of operations. The reason a fixed order exists at all is so that every reader interprets the same expression the same way. Simplify grouped parts first, handle exponents next, then multiplication and division left to right, then addition and subtraction left to right. With nested brackets, start with the innermost; a fraction bar also groups terms, so numerator and denominator stay together until simplified.

Worked example

Evaluate

First simplify the brackets:

Now evaluate the exponent:

Next do division and multiplication from left to right. Division comes first here because it appears first:

Then multiply:

Finish addition and subtraction from left to right:

So the value of the expression is $20$.

Try it yourself, then check the answer

Evaluate

Working in order: the bracket gives $5$, the exponent gives $9$, then $24/4 = 6$, then $6 \times 5 = 30$, and finally $30 - 9 = 21$. Write each stage on a new line and compare your intermediate lines, not just the final number.

Calculation traps to watch for

• Thinking division comes before multiplication. In $20 / 5 \times 2$, divide first because it appears first: $4 \times 2 = 8$, not $20 / 10 = 2$.
• Thinking addition comes before subtraction. In $10 - 3 + 1$, go left to right: $10 - 3 = 7$, then $7 + 1 = 8$.
• Skipping the rewrite step. Most order-of-operations errors are structure errors, not deep math errors. Rewriting the expression after each stage keeps signs, exponents, and grouped terms in place.

When the order of operations matters

This rule applies whenever one expression mixes operations: school arithmetic, algebra, spreadsheet formulas, calculator input, many science formulas. If only one type of operation appears, the rule does little work. It earns its keep when different kinds of operations sit together and you need one consistent interpretation.