BODMAS covers four kinds of brackets, not just one. 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 the places where students most often lose marks: the different kinds of brackets and the order they are removed in, the fraction bar that works as a hidden bracket, the word “of”, and how fractions and decimals behave once they are dropped into a longer expression. Every example is worked line by line and checked.
The “B” in BODMAS is often taught as if there is only one kind of bracket. In fact there are four kinds of grouping symbol, and long expressions often use several at once so that the grouping is easy to see. One rule is the same everywhere: always work from the innermost group outwards. The rest of this guide shows how to do that without slips.
| 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. It is a line drawn directly over the terms it groups, instead of symbols on either side. It still appears in three places: over the terms on the top or bottom of a fraction, over everything inside a square root sign, and over repeating digits in a decimal.
Many South Asian textbooks, including those used for school boards in Pakistan and India, teach a fixed order for mixed brackets. It matches how those books nest the symbols, with the bar innermost and the square brackets outermost:
| Order | Bracket |
|---|---|
| 1st | Bar (vinculum) |
| 2nd | Parentheses ( ) |
| 3rd | Curly brackets { } |
| 4th | Square brackets [ ] |
This is a convention, not a law of mathematics. Some textbooks in other countries nest the symbols differently, for example ( ) inside [ ] inside { }. What never changes is the idea underneath: the innermost group goes first. When you meet a nested expression, follow your own textbook’s order, and if the brackets are nested properly, “innermost first” gives the same answer.
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. (Without the bar, 8 + 4 − 2 would be 12 − 2 = 10 as well, but the bar matters when the operation in front is a minus: 8 − 4 − 2 = 8 − 2 = 6, whereas 8 − 4 − 2 = 2.)
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
Starting from the outside means you try to use numbers that have not been worked out yet.
The fix: always work from the innermost group outwards: bar first, then ( ), then { }, then [ ] last. In Example 2 the answer to (6 ÷ 2) is needed before { } can be finished, and { } is needed before [ ] can be finished. The square brackets are almost always the outermost layer, so they are resolved last.
A fraction’s horizontal bar does more than separate the numerator from the 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, and anything sitting outside the fraction stays separate.
(a) (3 + 5) ÷ 4 = 8 ÷ 4 = 2. Both 3 and 5 sit “on the bar”, so they are grouped together before dividing.
(b) 3 + 5 ÷ 4 = 3 + 1.25 = 4.25. Here only the 5 is over the bar, so only 5 ÷ 4 is grouped. The 3 stays separate.
(c) A fraction with sums on both levels: (6 + 2)/(5 − 1) means (6 + 2) ÷ (5 − 1) = 8 ÷ 4 = 2. Work out the top, work out the bottom, then divide.
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.
Fraction problems often use the word “of”, as in “half of 20”. It means multiply: 1/2 of 20 = 1/2 × 20 = 10. In many South Asian textbooks, the “O” in BODMAS stands for “of”, and “of” is worked out straight after the brackets, before division and multiplication. In the UK and many other places, the “O” stands for “orders” (powers and roots), and “of” is simply treated as a multiplication.
(a) 1/2 of 36 − 4: “of” first: 1/2 × 36 = 18. Then 18 − 4 = 14. Both conventions agree here.
(b) 8 ÷ 2 of 2: with the “of first” rule, 2 of 2 = 4, so 8 ÷ 4 = 2. With plain left-to-right division and multiplication, (8 ÷ 2) × 2 = 8.
The two conventions disagree on (b). The safe habit is to write brackets so nobody can misread it: 8 ÷ (2 of 2) or (8 ÷ 2) × 2. Follow the rule in your own syllabus in an exam, and use brackets in your own writing.
Once fractions appear inside a larger expression, BODMAS still applies exactly as before: brackets first, then multiplication and division, then addition and subtraction. The only difference is that you use fraction arithmetic at each step (see Fraction Fix-It and Multiply & Divide Fractions).
(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
2½ × (1/2 + 1/3)
Brackets first: 1/2 + 1/3 = 3/6 + 2/6 = 5/6
Change 2½ to 5/2, then multiply: 5/2 × 5/6 = 25/12 = 2 1/12
(3/4 ÷ 1/2) − 1/4
Brackets first: 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2
Then subtract: 3/2 − 1/4 = 6/4 − 1/4 = 5/4, which is 1¼
(1/2 + 1/3) / (1/6)
Top first: 1/2 + 1/3 = 5/6. The bar means divide, so 5/6 ÷ 1/6 = 5/6 × 6/1 = 5.
Decimals follow BODMAS exactly the same way. The only extra care needed is keeping the decimal point in the right place through each multiplication, division and addition step.
Two useful tricks: to multiply decimals, multiply as whole numbers and then count the decimal places (0.3 × 0.2: 3 × 2 = 6, and there are 2 decimal places in total, so 0.06). To divide by a decimal, multiply both numbers by 10 or 100 until the divisor is whole (6.4 ÷ 0.8 = 64 ÷ 8 = 8).
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 are not 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.
When a question mixes both, convert everything to one form first. Use decimals when they are short and exact, and fractions when a decimal would go on for ever.
(a) 1/4 + 0.5: change 1/4 to 0.25, so 0.25 + 0.5 = 0.75.
(b) 3/4 − 1/3: do not use 0.333…, which is never exact. Use fractions: 9/12 − 4/12 = 5/12.
(c) (1/2)² + 0.25 × 2: power first, (1/2)² = 1/4 = 0.25. Multiply: 0.25 × 2 = 0.5. Add: 0.25 + 0.5 = 0.75.
(d) √0.36 + 1/5: the root first, √0.36 = 0.6. Then 0.6 + 0.2 = 0.8.
Tap any expression below to see it solved step by step, including fraction and decimal expressions.
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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: Rs 4,500 − (0.20 × Rs 4,500) = Rs 4,500 − Rs 900 = Rs 3,600. Brackets, then multiplication, then subtraction, exactly as BODMAS requires.
A driver travels 3.5 hours at 60 km/h and then 1.25 hours at 80 km/h. What is the average speed for the whole trip?
Distance: 3.5 × 60 + 1.25 × 80 = 210 + 100 = 310 km.
Time: 3.5 + 1.25 = 4.75 hours.
Average speed = 310 ÷ (3.5 + 1.25) = 310 ÷ 4.75 ≈ 65.26 km/h. The brackets are essential. Without them, 310 ÷ 3.5 + 1.25 would give about 89.8, which is nonsense.
Notice that the average speed is not (60 + 80) ÷ 2 = 70 km/h, because more time was spent at the slower speed.
The vinculum bar is also used to mark repeating decimals: a bar drawn over the repeating digits, like 0.3 to mean 0.333… for ever. It is the same grouping symbol, used for a completely different purpose.
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1. Brackets: 1/3 + 1/6 = 2/6 + 1/6 = 1/2. Then 1/2 × 4 = 2.
2. Brackets: 0.9 + 0.3 = 1.2. Then 3.6 ÷ 1.2 = 3. Then 3 − 1 = 2.
3. The bottom of the fraction first: 2 + 4 = 6. Then 12 ÷ 6 = 2. Then 5 + 2 = 7.
1. Parentheses: 4 ÷ 2 = 2. Curly: 6 − 2 = 4. Square: 8 + 4 = 12. Then 24 − 12 = 12.
2. Brackets: 2/3 + 1/4 = 8/12 + 3/12 = 11/12. Then 11/12 × 6 = 66/12 = 11/2 = 5½.
3. Brackets: 4.2 − 1.8 = 2.4. Multiply: 0.5 × 2.4 = 1.2. Add: 1.5 + 1.2 = 2.7.
4. Distance = 3.5 × 60 + 1.25 × 80 = 210 + 100 = 310 km. Time = 3.5 + 1.25 = 4.75 h. Average speed = 310 ÷ 4.75 = 65.26 km/h (2 d.p.).
Ask learners to colour each bracket layer a different colour and to peel the expression from the inside, like an onion. Make them say aloud what is grouped by a fraction bar before they calculate. Tell them which bracket convention your syllabus uses, and also tell them that other books may differ, so that they are not surprised. Use real tasks (scaling a recipe, finding an average speed, working out a discount) so that the brackets have a reason to be there. Encourage one step per line, which is the most dependable way to cut errors.
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 usable everyday skill. It is also the direct preparation for algebra, where the same grouping rules apply to letters.
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