How do you use PEMDAS step by step?
With PEMDAS, you work through a calculation in four passes. Do whatever is inside parentheses first, then exponents, then multiplication and division from left to right, and last of all addition and subtraction from left to right. Rewrite the whole line after each pass, so any mistake is easy to spot later.
Example
Work out 3 + 4 × (6 − 2)² ÷ 8.
- Parentheses: 6 − 2 = 4, which leaves 3 + 4 × 4² ÷ 8
- Exponents: 4² = 16, which leaves 3 + 4 × 16 ÷ 8
- Multiplication and division, left to right: 4 × 16 = 64, then 64 ÷ 8 = 8, which leaves 3 + 8
- Addition and subtraction: 3 + 8 = 11
Copying the line out again feels slow. It's still the best protection against mistakes. A typical slip is to add the 3 and the 4 at the start. That gives 7, and the final answer is wrong.
Does multiplication come before division in PEMDAS?
No. Multiplication and division share the same level, and so do addition and subtraction. Inside each pair you simply go from left to right. PEMDAS lists M before D only because a word has to put its letters in some order. Treating the letters as a strict ranking is one of the most frequent order of operations mistakes.
Example
Work out 24 ÷ 4 × 2.
- The division comes first from the left: 24 ÷ 4 = 6
- Then the multiplication: 6 × 2 = 12
Multiply first and you get 24 ÷ 8 = 3, which is wrong. Addition and subtraction behave the same way. In 10 − 4 + 3, left to right gives 6 + 3 = 9. Adding first gives 10 − 7 = 3, also wrong.
The free textbook OpenStax Prealgebra 2e puts it plainly. Multiplication and division are done in order from left to right, because they have equal priority. Plenty of adults don't remember that detail from school, and it's behind many of the arguments about math puzzles online.
What is the difference between PEMDAS, BODMAS and BIDMAS?
There's no difference in the rules, only in the words. PEMDAS starts with P for parentheses, while BODMAS and BIDMAS start with B for brackets. E (exponents), O (orders) and I (indices) all mean powers and roots. Every version puts multiplication and division on one level, and addition and subtraction on the next.
Example
Work out 2 + 3 × 4² with each version in mind.
- Powers first: 4² = 16
- Then multiplication: 3 × 16 = 48
- Then addition: 2 + 48 = 50
BODMAS happens to write D before M, and PEMDAS writes M before D. Both give 50 here, and the two versions give the same answer in every calculation. If the letter order keeps causing confusion, it can help to write the rule as four levels instead of six letters: brackets, powers, multiply or divide, add or subtract.
Why do we need an order of operations?
Without an agreed order, a line like 2 + 3 × 4 could mean two different things. Add first and you get 20. Multiply first and you get 14. The convention says multiplication comes first, so the answer is 14. The point of the rule is simply that everyone reading the same calculation gets the same result.
Example
A friend buys 3 notebooks at 4 coins each and one pen for 2 coins. The total is 2 + 3 × 4 = 2 + 12 = 14 coins. Adding first would give 5 × 4 = 20, as if the pen had been bought four times over.
The convention is much older than the acronyms. According to MacTutor, the history of mathematics archive at the University of St Andrews, multiplication before addition and subtraction was already the rule in the earliest books of symbolic algebra in the 16th century. The same page quotes a historian writing in the late 1920s who said there was still no agreement on whether ÷ or × should come first.
What is the answer to 8 ÷ 2(2 + 2)?
If you follow the order of operations as taught, 8 ÷ 2(2 + 2) gives 16. The bracket comes first and gives 4. The 2 written next to the bracket means multiplication, and division and multiplication go from left to right, so 8 ÷ 2 × 4 = 4 × 4 = 16.
Example
Here are the two readings side by side.
- Left to right: 8 ÷ 2 × 4 = 4 × 4 = 16
- With 2(2 + 2) as one block: 8 ÷ (2 × 4) = 8 ÷ 8 = 1
The real problem is how the calculation is written. A note on the Berkeley mathematics department website points out that there is no standard convention for expressions like this, because writing a number right against a bracket suggests the two belong together. Mathematicians avoid the whole argument with extra brackets: (8 ÷ 2)(2 + 2) or 8 ÷ (2(2 + 2)).
A fraction bar groups things in the same way. If 6 + 4 is written above a fraction bar and 2 below it, you add first, so the answer is 10 ÷ 2 = 5.
What are the most common order of operations mistakes?
Most mistakes come from reading the acronym as six separate steps in a fixed order. That leads people to multiply before dividing or add before subtracting. Other slips are forgetting that a fraction bar groups what is above and below it, and squaring a negative number without its brackets. Rewriting the line after each step catches most of them.
The slips that keep coming back
- Multiplying before dividing. In 24 ÷ 4 × 2, go left to right: 6 × 2 = 12.
- Adding before subtracting. In 10 − 4 + 3, go left to right: 6 + 3 = 9.
- Squaring a negative number. −3² means −(3²) = −9, but (−3)² = 9. The bracket decides whether the minus sign gets squared.
- Starting with the outer bracket. In 2 × [5 + (3 − 1)²], begin inside: 3 − 1 = 2, then 2² = 4, then 5 + 4 = 9, and finally 2 × 9 = 18.
- Ignoring the fraction bar. Work out the top and the bottom on their own first, then divide.
How can you practise the order of operations?
Short, frequent calculations are the best way to practise this rule. In the order of operations game, each attack is a mixed calculation such as (5 + 3) × 7. You work out the bracket, multiply, and type 56 to land the attack.
Exponents slow a lot of people down, especially with negative numbers. If that is where you get stuck, try the exponents game first. After that, longer calculations with powers are easier to work through.
The name of the rule depends on where you learn it. In the US it's PEMDAS, and in the UK it's BODMAS or BIDMAS. Other languages use their own short phrases, but the rule underneath is the same everywhere.
One shared rule means everyone gets the same answer
The order of operations is an agreement, not a law of nature, and it only works because everyone uses the same one. Brackets first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. When a calculation could still be read two ways, add brackets and the argument is over.
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