TL;DR: GEMDAS stands for Grouping symbols, Exponents, Multiplication and Division, Addition and Subtraction, and it is the same rule the rest of the world calls PEMDAS or BODMAS. Philippine schools teach it in stages: MDAS in Grade 4, GMDAS or PMDAS in Grade 5, and GEMDAS by the end of Grade 6 once exponents appear, with Grade 7 applying it to integers. The acronym hides two rules that cause most errors: multiplication and division are one step done left to right, and addition and subtraction are one step done left to right. Grouping covers more than round brackets, including the fraction bar, the radical sign and absolute value bars.
If you searched for GEMDAS examples with answers, what follows is eleven of them, each solved line by line with the reason for every step, plus a table of the wrong answers these expressions usually produce and why. For the wider subject this sits in, see our DepEd mathematics curriculum guide.
GEMDAS, PEMDAS, BODMAS: Same Rule, Different Names
The order of operations is a convention, an agreement that everyone reads the same string of symbols the same way. Different countries built different mnemonics for it, and none of them describes a different mathematics.
| Acronym | Where it is common | First letters stand for |
|---|---|---|
| GEMDAS | Philippines | Grouping symbols, Exponents |
| PEMDAS | United States | Parentheses, Exponents |
| BODMAS | United Kingdom and much of the Commonwealth | Brackets, Orders |
| BIDMAS | United Kingdom, alternative form | Brackets, Indices |
| BEDMAS | Canada | Brackets, Exponents |
The last four letters are identical in all of them: Multiplication and Division, then Addition and Subtraction. So a Filipino learner reading a British worksheet and a British learner reading a Filipino one will get the same answers.
The reason for preferring G over P is practical. Parentheses are only one grouping device, and learners who memorise "parentheses first" often do not recognise that a fraction bar or a square root sign groups too.
| Grouping symbol | Example | What it groups |
|---|---|---|
| Parentheses | (4 + 2) | Everything between them |
| Brackets and braces | [ ] and { } | Usually used to nest around parentheses |
| Fraction bar (vinculum) | the bar in a fraction | The whole numerator and the whole denominator, separately |
| Radical sign | the square root symbol | Everything under the bar |
| Absolute value bars | | | | Everything between them |
How Philippine Schools Introduce It, Grade by Grade
The MATATAG mathematics curriculum builds the rule in stages rather than teaching the whole acronym at once, which is why a Grade 4 learner and a Grade 7 learner will name it differently and both be right.
| Level | What the curriculum expects | Name used |
|---|---|---|
| Grade 4 | Perform two or more different operations on whole numbers | MDAS rules |
| Grade 5 | Simplify a series of operations on whole numbers involving more than two operations, now with grouping symbols | PMDAS or GMDAS rule |
| End of Grade 6 | Exponential form, including calculation using the rules | GEMDAS rules |
| Grade 7 | Simplify numerical expressions involving integers using number properties and the order of operations | GEMDAS |
The E only appears once exponents do. That is the whole reason a Grade 5 worksheet says GMDAS and a Grade 7 worksheet says GEMDAS. For the phased rollout of this curriculum, see our MATATAG curriculum guide.
The Two Rules the Acronym Hides
Rule 1: multiplication and division are one step, worked left to right. The M sitting before the D in the acronym is an accident of spelling, not a ranking.
Rule 2: addition and subtraction are one step, worked left to right. Same reason.
Almost every wrong answer in the next section comes from ignoring one of these two.
Eleven Worked Examples
Example 1: MDAS with no grouping
8 + 6 × 3 − 4 ÷ 2
- Multiplication and division first, left to right: 6 × 3 = 18, then 4 ÷ 2 = 2
- The expression becomes: 8 + 18 − 2
- Addition and subtraction, left to right: 8 + 18 = 26, then 26 − 2 = 24
Answer: 24. Working strictly left to right instead would give 42 at the first step and a wrong final answer.
Example 2: division before multiplication
24 ÷ 4 × 2
- Equal rank, so left to right: 24 ÷ 4 = 6
- Then 6 × 2 = 12
Answer: 12. Doing the multiplication first gives 24 ÷ 8 = 3, which is the classic error.
Example 3: subtraction before addition
20 − 8 + 5
- Equal rank, so left to right: 20 − 8 = 12
- Then 12 + 5 = 17
Answer: 17. Doing the addition first gives 20 − 13 = 7, which is wrong for the same reason as Example 2.
Example 4: GMDAS with two groupings
5 × (12 − 8) + 36 ÷ (2 + 4)
- Grouping first, both of them: 12 − 8 = 4 and 2 + 4 = 6
- The expression becomes: 5 × 4 + 36 ÷ 6
- Multiplication and division, left to right: 5 × 4 = 20, then 36 ÷ 6 = 6
- Addition: 20 + 6 = 26
Answer: 26.
Example 5: nested grouping
3 × [18 − (4 + 2 × 3)] + 10
- Innermost grouping first, and the order of operations applies inside it too: 2 × 3 = 6, then 4 + 6 = 10
- The expression becomes: 3 × [18 − 10] + 10
- Remaining grouping: 18 − 10 = 8
- Multiplication: 3 × 8 = 24
- Addition: 24 + 10 = 34
Answer: 34. Note the step people skip: inside a bracket you do not simply work left to right, you apply the whole rule again.
Example 6: GEMDAS with an exponent
4 + 3 × 2³ − 10 ÷ 5
- Exponent first: 2³ = 8
- The expression becomes: 4 + 3 × 8 − 10 ÷ 5
- Multiplication and division, left to right: 3 × 8 = 24, then 10 ÷ 5 = 2
- Addition and subtraction, left to right: 4 + 24 = 28, then 28 − 2 = 26
Answer: 26.
Example 7: an exponent applied to a group
(6 − 2)² ÷ 8 + 7
- Grouping first: 6 − 2 = 4
- The expression becomes: 4² ÷ 8 + 7
- Exponent: 4² = 16
- Division: 16 ÷ 8 = 2
- Addition: 2 + 7 = 9
Answer: 9. Squaring before subtracting would give 36 − 4, a different expression entirely.
Example 8: the fraction bar as a grouping symbol
(15 + 9) over (2 × 3), written as a single fraction
- The bar groups the numerator and the denominator separately, so both are finished first: 15 + 9 = 24 and 2 × 3 = 6
- Then divide: 24 ÷ 6 = 4
Answer: 4. If you rewrite a fraction on one line, you must add brackets, because 15 + 9 ÷ 2 × 3 is a different expression with a different value.
Example 9: GEMDAS with integers
−5 + 3 × (−4) − (−6) ÷ 2
- Multiplication and division, left to right: 3 × (−4) = −12, then (−6) ÷ 2 = −3
- The expression becomes: −5 + (−12) − (−3)
- Addition and subtraction, left to right: −5 + (−12) = −17, then −17 − (−3) = −17 + 3 = −14
Answer: −14. Subtracting a negative is the step that trips learners here, not the ordering.
Example 10: where the negative sign sits
−3² compared with (−3)²
- In −3², the exponent attaches to the 3 only, so this reads as the negative of 3 squared: 3² = 9, and the answer is −9
- In (−3)², the grouping symbols make the negative number itself the base: (−3) × (−3) = 9
Answers: −9 and 9. Same digits, different values, and the only difference is a pair of brackets.
Example 11: a word problem
A class buys 4 boxes of 12 pencils, sets aside 15 pencils for a contest, and shares the rest equally among 3 groups. How many does each group get?
- Write the expression, using grouping to show what must happen first: (4 × 12 − 15) ÷ 3
- Inside the grouping, multiplication before subtraction: 4 × 12 = 48, then 48 − 15 = 33
- Division: 33 ÷ 3 = 11
Answer: 11 pencils per group. The grouping symbols are doing real work here: without them, 4 × 12 − 15 ÷ 3 means something the problem never said.
The Expression Everyone Argues About
6 ÷ 2(1 + 2)
Under the GEMDAS convention taught in Philippine schools:
- Grouping first: 1 + 2 = 3, giving 6 ÷ 2(3)
- The number written against a bracket means multiplication, so this is 6 ÷ 2 × 3
- Multiplication and division are equal rank, worked left to right: 6 ÷ 2 = 3, then 3 × 3 = 9
Answer under GEMDAS: 9.
The reason this circulates online is that a different convention exists. In some scientific and typesetting practice, multiplication written without a symbol, as in 2(3), is treated as binding more tightly than division, which turns the expression into 6 ÷ 6 and gives 1. Neither side is doing arithmetic wrong; they are reading the same marks under different notation rules. The professional response is that the expression is badly written, and the fix is a bracket: write either (6 ÷ 2)(1 + 2) or 6 ÷ [2(1 + 2)] and the argument disappears. Take the 9 for a Philippine classroom answer, and never write an expression like this yourself.
Wrong Answers and Where They Come From
| Expression | Common wrong answer | The mistake | Correct |
|---|---|---|---|
| 24 ÷ 4 × 2 | 3 | Multiplied first because M precedes D in the acronym | 12 |
| 20 − 8 + 5 | 7 | Added first because A precedes S in the acronym | 17 |
| 8 + 6 × 3 | 42 | Worked strictly left to right, ignoring rank | 26 |
| 2 + 3 × 4² | 400 | Worked left to right and applied the exponent last | 50 |
| −3² | 9 | Attached the exponent to the negative sign | −9 |
| 6 ÷ 2(1 + 2) | 1 | Treated 2(3) as a single grouped quantity | 9 under GEMDAS |
The fourth row worked in full: 4² = 16 first, then 3 × 16 = 48, then 2 + 48 = 50. The wrong answer of 400 comes from adding, multiplying and squaring in written order instead of by rank.
A Checking Routine That Catches Errors
- Rewrite the entire expression on a new line after every step. Most errors are copying errors, not reasoning errors, and this is the only habit that catches them.
- Do one rank per line. All the grouping, then all the exponents, then all the multiplication and division together, then all the addition and subtraction together.
- Remove a bracket only when what is inside it is a single number. A half-simplified bracket carried forward is a guaranteed error.
- Read left to right within a rank, out loud if necessary. Say "divide first because it comes first".
- Estimate before you finish. If 8 + 6 × 3 − 4 ÷ 2 is around 24, an answer of 42 should feel wrong before you check it.
- Add brackets to your own work. Nothing in the rules stops you writing (6 × 3) instead of 6 × 3. Redundant brackets cost nothing and prevent misreading.
Why the Rule Exists at All
The order of operations is not a discovered fact about numbers. It is a shared agreement that lets one written expression mean one thing to every reader, in the same way that spelling conventions let one written word mean one thing. That has a useful consequence for learners: grouping symbols always outrank convention. If you are unsure how an expression will be read, add brackets and the ambiguity is gone. For where this fits in the wider programme, see our K to 12 curriculum guide.
Disclaimer: curriculum competencies and their grade placement are set by DepEd and revised from time to time, and the MATATAG curriculum is being rolled out in phases. Confirm what applies to your grade level with your teacher or against the official curriculum guide before relying on this for graded work.
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Sources
- DepEd — MATATAG Mathematics Curriculum Guide, Grades 1 to 4 and 7 (official PDF)
- DepEd Learning Resource Portal — Grade 5 Mathematics, simplifying a series of operations using PMDAS or GMDAS
- DepEd — DO 010, s. 2024, Policy Guidelines on the Implementation of the MATATAG Curriculum
- SchoolFinderPH — DepEd Mathematics Curriculum Guide
- SchoolFinderPH — MATATAG Curriculum Guide
