3 + 4 × 2 is 11. Worked left to right it looks like 14, and that single habit is the most common source of arithmetic errors among people whose arithmetic is otherwise fine. The rule that fixes it has several names — BODMAS in the UK, BIDMAS in some schools, PEMDAS in the US — and they are all the same rule with different initials. What they do not make obvious is the part that actually trips people up: two of the letters are not a sequence at all.
Brackets — anything inside brackets first, working outwards from the innermost pair.
Orders — powers and roots. BIDMAS calls these Indices, PEMDAS calls them Exponents. Same thing.
Division and Multiplication — these rank equally, worked left to right.
Addition and Subtraction — these also rank equally, worked left to right.
BODMAS, BIDMAS and PEMDAS are identical. B and P both mean brackets or parentheses; O, I and E all mean powers. The only real difference is which of the equal-ranking pairs gets written first, and since they rank equally the order they are written in does not matter.
Two lines, not four. Brackets and powers first, then the times-and-divide line, then the plus-and-minus line — each line worked left to right.
Treating multiply and divide as a sequence
12 ÷ 3 × 2 is 8, not 2. Division and multiplication rank equally, so this is worked left to right: 12 ÷ 3 = 4, then 4 × 2 = 8. Doing the multiplication first because the M comes before the D in BODMAS gives 12 ÷ 6 = 2, which is wrong.
The same trap exists in PEMDAS, where the M comes first. Neither acronym means what it appears to mean at that point, which is the biggest weakness of teaching the rule as a word.
Treating add and subtract as a sequence
10 − 3 + 2 is 9, not 5. Again these rank equally and run left to right: 10 − 3 = 7, then 7 + 2 = 9. Doing the addition first gives 10 − 5 = 5.
A useful way to see it: subtraction is adding a negative. 10 − 3 + 2 is 10 + (−3) + 2, and additions can be done in any order, which lands on 9 from every direction.
3 + 4 × 2
Multiplication before addition, so 4 × 2 = 8 first, then 3 + 8 = 11.
(3 + 4) × 2
The brackets change everything. 3 + 4 = 7 first, then 7 × 2 = 14. This is what the brackets are for — they exist precisely to override the default order.
2 + 3² × 4
Powers before multiplication before addition. 3² = 9, then 9 × 4 = 36, then 2 + 36 = 38.
20 − 4 × 3 + 6 ÷ 2
Do the times-and-divide line first, left to right: 4 × 3 = 12 and 6 ÷ 2 = 3. That leaves 20 − 12 + 3, worked left to right: 8 + 3 = 11.
(8 − 3)² ÷ 5
Innermost brackets first: 8 − 3 = 5. Then the power: 5² = 25. Then the division: 25 ÷ 5 = 5.
It is a convention, not a law of arithmetic. Someone had to decide what 3 + 4 × 2 means, because written that way it is genuinely ambiguous, and the convention chosen makes algebra tidier: it lets 3x + 4 be written without brackets, which would otherwise be needed constantly.
That is also why the rule is universal. Every calculator, spreadsheet and programming language follows it, so a formula written one way gives the same answer everywhere. Where it appears to break down — the viral expressions that circulate with two defensible answers — the expression is almost always genuinely ambiguous, usually because of implicit multiplication next to a division sign.
The practical response to an ambiguous expression is not to argue about the convention. It is to add brackets.
Knowing the rule and applying it while hurrying are different skills, and the gap between them is where careless mistakes live. Three habits close it.
Mark what goes first. Underline or circle the part you will evaluate before you start. One second of marking prevents most of the errors.
Rewrite as you go. After each step, write the whole expression out again with that part replaced. Trying to hold a half-simplified expression in your head is what loses the minus signs.
Sanity-check the size. 3 + 4 × 2 being 11 rather than 14 is a small difference, but 2 + 3² × 4 being 38 rather than 100 is not. A rough estimate catches the structural mistakes.
Then drill it. Order of Operations gives you sixty seconds of expressions with the working shown on any you miss, which is the fastest way to make the habit automatic rather than remembered.
Brackets, Orders, Division, Multiplication, Addition, Subtraction. Orders means powers and roots. BIDMAS uses Indices and PEMDAS uses Parentheses and Exponents, but all three are the same rule.
Because multiplication is done before addition. 4 × 2 = 8 first, then 3 + 8 = 11. Working left to right would give 14, which is the classic mistake.
Both. They are the same rule with different initials — B and P both mean brackets, O and E both mean powers. Neither country does the arithmetic differently.
No. Multiplication and division rank equally and are worked left to right, so 12 ÷ 3 × 2 is 8, not 2. The same applies to addition and subtraction.
Brackets. Evaluate everything inside the brackets first, working outwards from the innermost pair, then apply powers.
Because without it an expression like 3 + 4 × 2 has two defensible answers. The convention makes sure a formula means the same thing to every person, calculator and spreadsheet.
Brackets, then powers, then the times-and-divide line left to right, then the plus-and-minus line left to right. Two of those lines are pairs of equals, which is the part the acronym hides. Drill it on [Order of Operations](/games/order-of-operations) until marking the first step becomes automatic.