BODMAS covers four kinds of brackets, not just one — and once fractions and decimals join an expression, a few extra habits become essential. This guide fills in both gaps.
Our BODMAS & DMAS Rules article covered the core order of operations. This companion guide goes deeper into two things that article only touched lightly: the four different kinds of brackets and the order they're removed in, and how fractions and decimals behave once they're dropped into a BODMAS expression.
The "B" in BODMAS is often taught as if there's only one kind of bracket. In fact, there are four, and textbooks that use several types in the same expression expect them to be removed in a specific, agreed order — not just "whichever looks innermost."
| Bracket | Symbol | Also Called |
|---|---|---|
| Bar | 4 − 2 | Vinculum, line bracket |
| Parentheses | ( ) | Round brackets, small brackets |
| Curly Brackets | { } | Braces |
| Square Brackets | [ ] | Box brackets |
The bar (vinculum) is the oldest and least common today — a line drawn directly over the terms it groups, instead of symbols on either side. It still appears in specific places: grouping a numerator over a fraction line, or marking a repeating decimal.
When an expression mixes several bracket types together, the standard convention removes them in this fixed order:
| Order | Bracket |
|---|---|
| 1st | Bar (vinculum) |
| 2nd | Parentheses ( ) |
| 3rd | Curly brackets { } |
| 4th | Square brackets [ ] |
In practice, this order usually lines up with "innermost first" anyway, since textbooks nest brackets in exactly this sequence — square brackets on the outside, then curly, then parentheses, then a bar innermost of all.
8 + 4 − 2: the bar groups "4 − 2" as if it were in its own brackets. Resolve the bar first: 4 − 2 = 2. Then: 8 + 2 = 10.
100 − [20 + {10 − (6 ÷ 2)}] ÷ 2
Step 1 — parentheses (innermost): 6 ÷ 2 = 3
Step 2 — curly brackets: 10 − 3 = 7
Step 3 — square brackets: 20 + 7 = 27
Step 4 — division: 27 ÷ 2 = 13.5
Step 5 — subtraction: 100 − 13.5 = 86.5
The removal order has nothing to do with alphabetical order or which symbol looks more important — it follows the fixed bar → parentheses → curly → square sequence.
The fix: always work from the innermost bracket type outward: bar first, then ( ), then { }, then [ ] last. Square brackets are almost always the outermost layer, resolved last.
A fraction's horizontal bar does more than separate numerator from denominator — it silently groups everything written above it and everything written below it, exactly like a bracket. Only terms actually written on the fraction bar are grouped; anything sitting outside the fraction stays separate.
(3+5) ÷ 4 = 8 ÷ 4 = 2 — both 3 and 5 sit "on the bar," so they're grouped together before dividing.
Compare: 3 + 5÷4 = 3 + 1.25 = 4.25 — here, only the 5 is written over the fraction bar, so only 5÷4 is grouped; the 3 stays separate.
Without brackets, only the term actually written as part of the fraction gets grouped by the fraction bar.
The fix: 3 + 5/4 means 3 + (5÷4) = 3 + 1.25 = 4.25. If both 3 and 5 were meant to be added before dividing by 4, the expression would need explicit brackets: (3+5)/4 = 2. Always check exactly what sits directly above and below the fraction bar.
Once fractions appear inside a larger expression, BODMAS still applies exactly as before — brackets first, then multiplication/division, then addition/subtraction — just using fraction arithmetic at each step instead of whole numbers.
(1/2 + 1/4) × 8
Brackets first, common denominator: 1/2 + 1/4 = 2/4 + 1/4 = 3/4
Then multiply: 3/4 × 8 = 6
3/4 × (2/3 + 1/6)
Brackets first: 2/3 + 1/6 = 4/6 + 1/6 = 5/6
Then multiply: 3/4 × 5/6 = 15/24 = 5/8
2/3 + 1/3 × 3/5
Multiply first: 1/3 × 3/5 = 3/15 = 1/5
Then add: 2/3 + 1/5 = 10/15 + 3/15 = 13/15
Decimals follow BODMAS exactly the same way. The only extra care needed is keeping the decimal point aligned correctly through each multiplication, division, and addition step.
2.5 + 3 × 0.4
Multiply first: 3 × 0.4 = 1.2
Then add: 2.5 + 1.2 = 3.7
2.5 + 3 × 0.4 − (1.2 ÷ 0.6)
Brackets first: 1.2 ÷ 0.6 = 2
Multiply: 3 × 0.4 = 1.2
Left to right: 2.5 + 1.2 = 3.7, then 3.7 − 2 = 1.7
0.5 × (3.6 + 2.4) − 1.8
Brackets first: 3.6 + 2.4 = 6.0
Multiply: 0.5 × 6.0 = 3.0
Subtract: 3.0 − 1.8 = 1.2
Grouping terms that aren't actually in brackets is one of the most common decimal-BODMAS mistakes.
The fix: with no brackets present, division and multiplication are resolved left to right as separate steps: 6.4 ÷ 0.8 = 8, and 1.5 × 2 = 3. Then add: 8 + 3 = 11. Never combine terms across an operation unless brackets say so.
Tap any expression below to see it solved step by step, including fraction and decimal expressions.
Tap an expression above to see the full step-by-step solution.
A recipe calls for (3/4 + 1/2) cups of flour per batch, tripled for a party: (3/4 + 1/2) × 3 = (5/4) × 3 = 15/4 = 3¾ cups. A shopping receipt calculating a discount uses decimals the same way: $45.00 − (0.20 × $45.00) = $45.00 − $9.00 = $36.00 — brackets, then multiplication, then subtraction, exactly as BODMAS requires.
The vinculum bar is also used to mark repeating decimals — a bar drawn over the repeating digits, like 0.3 to mean 0.333... forever. It's the same grouping symbol, used for a completely different purpose.
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Real formulas — in recipes, finance, engineering, and science — almost never stick to whole numbers or a single bracket type. Being fluent with mixed brackets, fractions, and decimals together is what turns BODMAS from a classroom rule into a genuinely usable, everyday skill.
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