Order of Operations (PEMDAS) Explained
Why the order matters, and how to remember it
I once watched my son solve 2 + 3 × 4 on a worksheet. He wrote 20. "Two plus three is five," he said, "and five times four is twenty." The arithmetic was correct — every single step. The answer was wrong. The reason is that math has rules about which operation to do first, and without those rules, the same expression could mean different things to different people. This is what order of operations solves, and PEMDAS is the most common way to remember the sequence.
Why Order Matters
Consider the expression 8 − 2 × 3. Do the subtraction first and you get 6 × 3 = 18. Do the multiplication first and you get 8 − 6 = 2. Two very different answers from the same expression. Without an agreed order, math would be ambiguous. Computers, calculators, and mathematicians all use the same convention: multiplication and division are done before addition and subtraction. So 8 − 2 × 3 = 2, not 18.
This is not a trick or a cultural preference — it is a universal agreement that keeps every calculation consistent across the world. Once a child understands why the order exists (to avoid confusion), the rules stop being arbitrary and start making sense.
PEMDAS: The Memory Helper
PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. It is a mnemonic — a memory trick — for the correct sequence. Each letter stands for one type of operation, and you follow them from left to right.
P — Parentheses (brackets) first
E — Exponents (powers) next
M/D — Multiplication and Division (left to right)
A/S — Addition and Subtraction (left to right)
The most important thing to notice is that multiplication and division are equal, and so are addition and subtraction. You do not always multiply before divide, and you do not always add before subtract. You do whichever one appears first, working from left to right. This is where many students go wrong, and it is worth repeating.
Worked Example: Step by Step
Let us solve the expression my son got wrong: 2 + 3 × 4.
Step 1: Are there parentheses? No.
Step 2: Are there exponents? No.
Step 3: Multiplication or division? Yes — 3 × 4 = 12.
Step 4: Addition or subtraction? 2 + 12 = 14.
The answer is 14, not 20. Multiplication goes first because M comes before A in PEMDAS. This simple, step-by-step approach works every time, no matter how long the expression gets.
A Longer Example: Putting It All Together
Try this one: 6 + 2 × (3 + 1)² ÷ 4.
Step 1 — Parentheses: 3 + 1 = 4 → 6 + 2 × 4² ÷ 4
Step 2 — Exponents: 4² = 16 → 6 + 2 × 16 ÷ 4
Step 3 — Multiply/Divide left to right:
2 × 16 = 32 → 6 + 32 ÷ 4
32 ÷ 4 = 8 → 6 + 8
Step 4 — Add/Subtract: 6 + 8 = 14
Every step follows the same rule, just applied to a shorter and shorter expression. When children see PEMDAS as a checklist they run through once per step, the whole process becomes methodical rather than scary.
The Left-to-Right Trap
The biggest mistake students make with PEMDAS is always multiplying before dividing, regardless of position. Consider: 12 ÷ 3 × 2.
Wrong way: 3 × 2 = 6, then 12 ÷ 6 = 2.
Right way: 12 ÷ 3 = 4, then 4 × 2 = 8.
Division comes first because it is to the left. M and D are equal — you simply go left to right. The same rule applies to addition and subtraction. 7 − 3 + 2 is not 7 − 5; it is 4 + 2 = 6. Left to right, always.
I have seen this exact mistake on countless homework assignments. Once a child learns that "M/D are partners, A/S are partners, and left wins ties," the error almost never comes back.
Why PEMDAS Helps in Everyday Life
Order of operations is not just a school exercise. Imagine you are shopping: you buy 3 shirts at $12 each, and a $5 discount applies. The expression is 3 × 12 − 5 = 31, not 3 × 7 = 21. Multiplication goes first because you are multiplying the price by quantity.
Or consider a recipe: you double a recipe (× 2) and then subtract 1 cup of sugar (− 1). You must multiply the entire recipe first, then subtract. Doing it the other way would change the result. Real-life problems follow PEMDAS naturally — knowing the rule just makes the process visible.
I once split a restaurant bill with a friend: the total was $45 for two people, and we had a $6 coupon. The right way to think about it is $45 ÷ 2 − $6 = $16.50 per person. If we subtracted the coupon first, we would get $39 ÷ 2 = $19.50, which overcharges each of us by three dollars. That three-dollar difference is exactly what happens when PEMDAS is ignored. Small mistakes in order become real money.
These everyday moments are excellent teaching opportunities. When a child sees that the same rule that governs 2 + 3 × 4 also governs the bill at dinner, the abstraction becomes personal. And when they catch the mistake before you do, they own the knowledge in a way no worksheet can give.
Tips for Parents: Making PEMDAS Stick
Some children memorize PEMDAS quickly, while others need repeated practice with concrete examples. Here are a few strategies that help:
- Write the checklist. Have your child write P-E-M/D-A/S down the left side of the page before solving any problem. Each step becomes a row to check off.
- Use colour. Highlight multiplication and division in one colour, addition and subtraction in another. Visual distinction helps children see which group they are working in.
- Verbalize the left-to-right rule. "M and D are partners, A and S are partners, and left wins ties." Saying it out loud while solving makes the rule stick faster than silent reading.
- Catch mistakes on purpose. Write a wrong answer on purpose (do the addition before multiplication) and ask your child to find the error. Teaching someone else is the deepest form of learning.
The goal is not to rush through PEMDAS but to build the habit of scanning for parentheses, exponents, and operations before writing a single number. That two-second scan saves minutes of reworking later.
Frequently Asked Questions
Is PEMDAS the only order of operations?
No. Different countries use different mnemonics (BODMAS, BEDMAS, GEMS). They all describe the same rules — just with different letters for the same concepts.
Does multiplication always come before division?
No. They are equal in priority. You do whichever one appears first when reading the expression left to right.
What about exponents — are they always after parentheses?
Yes. Exponents are evaluated after anything inside parentheses but before multiplication, division, addition, and subtraction.
Why is PEMDAS so hard to remember?
It is not the letters that are hard — it is the "left to right" rule for equal partners. Once children practice that rule, PEMDAS becomes automatic.
Can calculators make PEMDAS mistakes?
Basic calculators follow left-to-right without PEMDOS. Scientific calculators follow PEMDAS correctly. This is why students should understand the rules before relying on a calculator.
Quick Practice Problems
- What is 5 + 3 × 2?
Answer: 3 × 2 = 6, then 5 + 6 = 11.
- What is 10 − 4 + 2?
Answer: Left to right: 10 − 4 = 6, then 6 + 2 = 8.
- Solve: 2 × (3 + 4).
Answer: Parentheses first: 3 + 4 = 7, then 2 × 7 = 14.
- What is 12 ÷ 4 × 3?
Answer: Left to right: 12 ÷ 4 = 3, then 3 × 3 = 9.
- Solve: 1 + 2².
Answer: Exponents first: 2² = 4, then 1 + 4 = 5.