Order of Operations (PEMDAS)

Why We Need an Agreed Order

Imagine two students are handed the same problem: \(3 + 4 \times 2\). The first student works from left to right, adding \(3 + 4 = 7\), then multiplying \(7 \times 2 = 14\). The second student multiplies first, \(4 \times 2 = 8\), then adds \(3 + 8 = 11\). Same problem, two different answers — 11 and 14. They can't both be right.

Math is a language, and like any language it needs shared rules so that everyone reads a sentence the same way. If \(3 + 4 \times 2\) could mean either 11 or 14 depending on who's reading it, the expression would be useless — a calculator in New York and a calculator in Tokyo might disagree. To prevent this, mathematicians long ago agreed on one universal set of rules for the order in which operations are carried out. Follow the rules, and the answer is always the same, everywhere, for everyone.

These rules are usually taught in grades 5 and 6, right before students step into algebra. That timing is no accident: once letters and variables enter the picture, knowing exactly which operation happens first becomes essential. Getting comfortable with the order of operations now makes the transition to algebra far smoother later.

PEMDAS: The Order of Operations

The rules are captured by a well-known mnemonic: PEMDAS. Each letter stands for a stage of evaluation, and you work through the stages in order from top to bottom:

1. P — Parentheses: Do everything inside grouping symbols first.
2. E — Exponents: Evaluate any powers (like \(3^2\)) next.
3. MD — Multiplication and Division: Handle these together, working left to right.
4. AS — Addition and Subtraction: Handle these together, working left to right.

Many students memorize PEMDAS with the phrase "Please Excuse My Dear Aunt Sally." That's a handy hook — but it hides a trap, and it's the single most important thing to understand about the order of operations.

In other words, the "M" is not more powerful than the "D." They are equal partners, and whichever one comes first as you read the expression from left to right gets done first. The same is true for the "A" and the "S." This is where a huge fraction of order-of-operations mistakes come from, so it's worth saying plainly: multiplication does not automatically beat division, and addition does not automatically beat subtraction.

Other Names for the Same Idea

You may see the same rules under different names depending on where you learned math:

• BODMAS / BIDMAS (common in the UK and elsewhere): Brackets, Orders (or Indices), Division, Multiplication, Addition, Subtraction.
• GEMS: Grouping, Exponents, Multiplication/division, Subtraction/addition.

Notice that BODMAS lists Division before Multiplication, while PEMDAS lists Multiplication before Division. This does not mean the two systems disagree! Both are just reminding you that those operations sit at the same level. Whichever name you use, the underlying rule is identical: same-rank operations are done left to right. Some teachers prefer GEMS precisely because it avoids the false impression that M beats D or A beats S.

Worked Example 1: \(3 + 4 \times 2\)

Let's settle the argument from the opening. Which comes first, the addition or the multiplication?

Step 1 — Scan for Each Stage

There are no parentheses and no exponents here. That takes us straight to Multiplication and Division. We have one multiplication, \(4 \times 2\), so it goes first — before the addition.

Step 2 — Do the Multiplication

Step 3 — Do the Addition

The correct answer is \(3 + 4 \times 2 = \boldsymbol{11}\). The student who got 14 fell into the "left to right no matter what" trap — they added before multiplying, which breaks the rules.

Worked Example 2: \((3 + 4) \times 2\)

Now watch what happens when we wrap the addition in parentheses. This is the whole reason parentheses exist: they let us override the normal order and force something to happen first.

Step 1 — Do What's Inside the Parentheses

Parentheses are the very first stage of PEMDAS, so we evaluate the inside before anything else.

Step 2 — Do the Multiplication

So \((3 + 4) \times 2 = \boldsymbol{14}\). Compare the two examples side by side: with no parentheses, \(3 + 4 \times 2 = 11\); with parentheses around the addition, \((3 + 4) \times 2 = 14\). The numbers and operations are identical — only the grouping changed — yet the answers differ. That is the power of parentheses, and it shows exactly why the order of operations matters.

Worked Example 3: \(2 + 6 \div 2 \times 3\)

This example is the real test of whether you've understood the "same rank, left to right" rule. There's a division and a multiplication sitting side by side. A student who believes "M always comes before D" will get this wrong.

Step 1 — Identify the Stages

No parentheses, no exponents. We move to Multiplication and Division. Here's the key question: we have both a division (\(6 \div 2\)) and a multiplication (\(\times 3\)). Which do we do first?

Because multiplication and division are equal in rank, we do not automatically pick the multiplication. Instead, we read left to right and do whichever appears first. Scanning from the left, the division \(6 \div 2\) comes before the multiplication, so the division happens first.

Step 2 — Do the Division (it's furthest left)

Step 3 — Do the Multiplication

Step 4 — Do the Addition

So \(2 + 6 \div 2 \times 3 = \boldsymbol{11}\). If you had insisted on multiplying first (\(2 \times 3 = 6\), then \(6 \div 6 = 1\), then \(2 + 1 = 3\)), you'd have gotten 3 — a completely different, and incorrect, answer. Left to right is not optional.

Worked Example 4: \(5 + 2 \times 3^2\)

Our last example brings in an exponent. Exponents live at the second stage of PEMDAS, above both multiplication and addition, so they get evaluated early.

Step 1 — Do the Exponent First

The expression \(3^2\) means "3 raised to the power of 2," which is \(3 \times 3\). Exponents come before multiplication and addition, so this is our first move.

Step 2 — Do the Multiplication

Step 3 — Do the Addition

So \(5 + 2 \times 3^2 = \boldsymbol{23}\). A common slip here is to multiply \(2 \times 3\) first and then square, or to square only after multiplying — both break the exponent-before-multiplication rule. The exponent attaches to the 3 alone, and it gets evaluated before anything else touches it.

Common Mistakes to Watch For

Almost every order-of-operations error falls into one of a few predictable patterns. Knowing them ahead of time is the best way to avoid them:

• Always doing multiplication before division: This is the most frequent mistake, and it comes straight from misreading PEMDAS. Multiplication and division are equal in rank — do them left to right. In \(6 \div 2 \times 3\), the division comes first only because it's written first.
• Ignoring precedence and working strictly left to right: Some students treat every problem like a straight left-to-right calculation, which turns \(3 + 4 \times 2\) into 14 instead of 11. Left to right applies only to operations that are tied in rank, not across different stages.
• Forgetting that exponents come before multiply and divide: In \(2 \times 3^2\), the square must be done first (\(3^2 = 9\), then \(2 \times 9 = 18\)). Multiplying before squaring gives the wrong answer.
• Treating subtraction as higher priority than addition: Just like M and D, addition and subtraction share a rank. In \(10 - 4 + 2\), you subtract first (because it's on the left), getting \(6 + 2 = 8\) — not \(10 - 6 = 4\). Don't let the subtraction jump the line.
• Ignoring or misplacing parentheses: Dropping the parentheses in \((3 + 4) \times 2\) changes the answer from 14 to 11. Whatever is grouped must be finished before it interacts with anything outside the group.

Tips for Parents and Teachers

• Underline or box the next step: Before doing anything, have the student underline (or draw a box around) the exact part they're about to evaluate. Physically marking "this is what happens next" slows them down just enough to apply the rules correctly. You'll see this technique used in the worked examples above.
• Say the stages out loud: Have the student recite "Parentheses, Exponents, Multiply-Divide left to right, Add-Subtract left to right" and point to each stage as they scan the problem. Verbalizing builds the habit.
• Emphasize "same rank" with a demonstration: Write \(8 \div 4 \times 2\) and \(8 \times 4 \div 2\) side by side and work both left to right. Seeing that the order genuinely depends on position — not on which symbol it is — makes the rule stick.
• Point to the viral math problems online: Those "90% of people get this wrong!" posts that flood social media almost always come from ambiguous or tricky order-of-operations expressions. They're a fun, real-world hook — and a chance to show your student that they now know the rules that the arguing commenters have forgotten.
• Practice little and often: A handful of expressions each day beats a long session once a week. Try a quick round of the Math Minute game or a short worksheet to keep the skill fresh and build speed.

Where This Leads

The order of operations is the grammar of arithmetic. Just as grammar tells you how to read a sentence so it means one clear thing, PEMDAS tells you how to read a mathematical expression so it has exactly one correct value. Once a student can confidently evaluate expressions with parentheses, exponents, and mixed operations — always applying "left to right" to tied ranks — they're ready to move on.

And the next stop is a big one: algebra. Every equation a student will ever solve, every formula they'll ever plug numbers into, depends on knowing which operation happens first. A student who has truly mastered the order of operations walks into algebra with a genuine advantage. When your child can breeze through problems like \(2 + 6 \div 2 \times 3\) and \(5 + 2 \times 3^2\) without hesitation — and can explain why each step comes when it does — they're ready for what comes next.