PEMDAS vs GEMDAS vs BODMAS
Three names, one set of rules. They differ in the words they use, never in the answer.
| Step | PEMDAS (US) | GEMDAS (US, newer) | BODMAS (UK, India, Australia) |
|---|---|---|---|
| 1 | Parentheses | Grouping symbols ( ) [ ] { } | Brackets |
| 2 | Exponents | Exponents | Orders (powers, roots) |
| 3 | Multiplication and division, left to right | Multiplication and division, left to right | Division and multiplication, left to right |
| 4 | Addition and subtraction, left to right | Addition and subtraction, left to right | Addition and subtraction, left to right |
Why they always agree
Each acronym lists the same four levels in the same order. The differences are vocabulary (parentheses, grouping symbols, brackets; exponents, orders) and which of two equal partners is written first (MD or DM). Since partners on the same level are worked left to right, the letter order inside a level never matters.
One problem, three sets of labels
The problem [122 ÷ (12 − 14)] + (−2) solved once. The steps are identical; only the label changes.
| # | PEMDAS | GEMDAS | BODMAS | Expression after the step |
|---|---|---|---|---|
| 1 | Parentheses | Grouping | Brackets | [122 ÷ (−2)] + (−2) |
| 2 | Parentheses, exponents | Grouping, exponents | Brackets, orders | [144 ÷ (−2)] + (−2) |
| 3 | Parentheses | Grouping | Brackets | −72 + (−2) |
| 4 | Add | Add | Addition | −74 |
Answer: −74.
Common mistakes
Multiplying before dividing
PEMDAS puts M before D, so students sometimes multiply first. 24 ÷ 4 × 2 is 6 × 2 = 12, working left to right, not 24 ÷ 8 = 3. BODMAS students make the mirror mistake and divide first.
Adding before subtracting
10 − 4 + 3 is 6 + 3 = 9, not 10 − 7 = 3. Addition and subtraction are one level, left to right.
A negative base with an exponent
−3² is −9 and (−3)² is 9. The exponent belongs to whatever is directly in front of it. Sheets here always put a negative base in parentheses.
Skipping the inner bracket
In nested grouping such as 2 × [10 − (3 + 4)], start with the innermost pair: 3 + 4, then 10 − 7, then 2 × 3.
More worked examples
- [72 × (6 − 2)] + 9 = 205
- Grouping: 6 − 2 = 4[72 × 4] + 9
- Grouping, exponents: 7² = 49[49 × 4] + 9
- Grouping: 49 × 4 = 196196 + 9
- Add: 196 + 9 = 205205
- 20 + (43 ÷ 2) × 11 = 372
- Grouping, exponents: 4³ = 6420 + (64 ÷ 2) × 11
- Grouping: 64 ÷ 2 = 3220 + 32 × 11
- Multiply: 32 × 11 = 35220 + 352
- Add: 20 + 352 = 372372
More worksheets
Questions
Do PEMDAS, GEMDAS and BODMAS give different answers?
No. They are three names for the same rules. PEMDAS says parentheses, GEMDAS says grouping symbols, BODMAS says brackets and orders. Every problem on this site has one answer whichever name you pick; only the words in the instructions and the answer-key labels change.
In PEMDAS, does multiplication come before division?
No. Multiplication and division share one step and are done from left to right, whichever comes first. The same goes for addition and subtraction. So 24 ÷ 4 × 2 is 12, not 3. BODMAS lists D before M for the same reason PEMDAS lists M before D: a mnemonic needs some order.
Is −3² equal to 9 or −9?
By the usual convention −3² means −(3²), which is −9, because the exponent applies before the minus sign. (−3)² is 9. Worksheets on this site always write a negative base in parentheses, such as (−3)², so the question never comes up on a sheet.
Which one should I teach?
Use the one your curriculum or textbook uses, so students see the same words in class and on homework. The generator can make the same sheet in any of the three.
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