The rule, precisely

In Discover you met the rule: brackets first, then × and ÷, then + and −, with partners going left to right. This layer makes that rule precise, shows you why each part of it is there, and gives you methods that work on long, messy expressions without mistakes.

You will learn to split an expression into terms, meet all the memory words (DMAS, BODMAS, BIDMAS, PEMDAS) and the trap hidden in each, use three kinds of brackets, handle the word of, turn word problems into expressions, and act as a checker who finds and fixes other people's mistakes. Along the way you will see why a cheap calculator and a scientific calculator can disagree.

An arithmetic expression is numbers joined by operation signs, possibly with brackets. It has a single value, the number you get when you work it out (we also say you simplify or evaluate it).

The + and − signs that are not inside brackets split an expression into pieces called terms. Each term is made only of numbers joined by × and ÷ (or a bracket).

Take 12 + 18 ÷ 3 × 2 − 4 × 5. Its terms are:

• 12
• 18 ÷ 3 × 2
• 4 × 5 (with a − in front of it)

This gives a neat method: work out each term on its own, then add and subtract the terms from left to right.

• Term 2: 18 ÷ 3 × 2 = 6 × 2 = 12
• Term 3: 4 × 5 = 20
• Put together: 12 + 12 − 20 = 4.

Different countries and textbooks remember the order of operations with different words. They all describe the same rule.

The important thing all of them hide: the letters are written in a line, but the rule has only three levels (four once powers appear in later classes):

1. Brackets (and powers such as 3², which you will meet properly in Class 7 and 8).
2. × and ÷, equal rank, left to right. ("Of" belongs here too, done just before the others.)
3. + and −, equal rank, left to right.

So DMAS does not mean "all divisions, then all multiplications, then all additions, then all subtractions". It means "÷ and × together, then + and − together".

Why must partners go left to right? Because − and ÷ belong to the number right after them.

In 10 − 3 + 2, the − sign says "take away 3". It does not say "take away 3 + 2". The 2 is being added. So the story is: start with 10, take away 3, add 2. That gives 9.

In 8 ÷ 4 × 2, the ÷ sign says "divide by 4". It does not say "divide by 4 × 2". The story is: start with 8, share into 4 groups (2 each), then double: 4.

For + and × alone, order never matters: 2 + 3 + 4 and 4 + 3 + 2 are both 9. The trouble only comes with − and ÷, because 10 − 3 is not 3 − 10 and 8 ÷ 4 is not 4 ÷ 8. Going left to right keeps every − and ÷ attached to the correct number.

When one bracket sits inside another, it gets hard to see which closing bracket belongs to which opening one. So people use different shapes:

• ( ) round brackets (also called parentheses or small brackets)
• { } curly brackets (also called braces)
• ☐ square brackets (also called box brackets)

In many Indian textbooks the usual nesting is round inside curly inside square: [ { ( ) } ]. But the shapes do not have a ranking of their own. The rule is simply:

Work out the innermost brackets first, then move outwards.

Whatever shape the innermost bracket has, it goes first.

The O in BODMAS is read two ways. Indian and British school books usually take it to mean of, as in "half of 10" or "one-third of 27"; other books (and sites such as Math is Fun) read it as Orders, meaning powers and roots. "Of" means multiply: ½ of 10 = ½ × 10 = 5.

So why does "of" get its own letter? Because BODMAS-style textbooks follow a convention: "of" is done before ordinary × and ÷. The idea is that "½ of 16" is one tightly-bound quantity (8), not two separate pieces. This is a textbook agreement, not a law of mathematics — plenty of books and every programming language simply treat "of" as an ordinary ×.

• 40 ÷ ½ of 16 (BODMAS convention): first ½ of 16 = 8, then 40 ÷ 8 = 5.
• 40 ÷ ½ × 16 (with × instead of "of"): left to right, 40 ÷ ½ = 80, then 80 × 16 = 1,280.

That is a huge difference from one small word — which is exactly why most mathematicians avoid writing "÷ ½ of" at all and use brackets instead: 40 ÷ (½ × 16) says the same as the first line with no room for doubt.

This site follows the BODMAS convention whenever the word "of" appears, and says so in the question.

Turning a story into a single expression is a skill in its own right. It forces you to decide what is multiplied by what and what must be worked out first, which is exactly what the order of operations encodes.

A reliable method:

1. Find the chunks. Each "number of things × price (or size) of one" is a chunk. Write each chunk as a product.
2. Join the chunks with + or −.
3. Ask: is a whole total being taken away, shared or multiplied? If yes, wrap that total in brackets.
4. Evaluate using the order of operations.
5. Check that the answer makes sense (size, units).

Most wrong answers in this topic come from a small number of mistakes. If you know them by name, you can check your own work quickly.

1. Reading like a sentence. Doing operations strictly left to right, ignoring × ÷ first. 5 + 5 × 5 becomes 50 instead of 30.
2. The letter trap. Doing + before − or ÷ before × because of the order of letters. 10 − 3 + 2 becomes 5 instead of 9.
3. Dropping brackets. Losing a bracket halfway through, especially after a − sign. 20 − (5 + 3) becomes 20 − 5 + 3 = 18 instead of 12.
4. Doing two steps at once in your head and mixing them up. Write one step per line.
5. Wrong brackets in a word problem. Taking away only part of a bill, or sharing only part of an amount.

Machines have to be told the rule too, and not all of them are.

• Basic (four-function) calculators usually use immediate execution: each time you press an operation key, they finish the previous step. So 2 + 3 × 4 = shows 20. They are built for running totals (adding up a bill item by item).
• Scientific calculators and most phone calculator apps wait until you press = and then follow the order of operations: 2 + 3 × 4 = shows 14.
• Spreadsheets (like Excel or Google Sheets) use * for × and / for ÷. Typing =2+3*4 gives 14, and =(2+3)*4 gives 20.
• Programming languages such as Python and Scratch follow the order of operations too: 2 + 3 * 4 is 14.

To get the right answer from a basic calculator, work out the × and ÷ chunks first, or use its memory keys: M+ adds the display to memory, M− subtracts it, MR recalls the total.

Good mathematicians check. Three quick checks for this topic:

1. Estimate first. 38 × 5 + 97 is about 40 × 5 + 100 = 300. The exact answer, 287, is close. If you had got 3,876 (by adding 5 + 97 first), the estimate would tell you at once that something went wrong.
2. Put the brackets in yourself. Rewrite the expression with brackets around every × and ÷ chunk: 38 × 5 + 97 becomes (38 × 5) + 97. Then there is only one way to read it.
3. Use meaning. If the expression came from a story, does the answer fit the story? Change cannot be more than the note you paid with; a share cannot be bigger than the whole.

Shops, schools and offices keep bills in spreadsheets such as Excel, Google Sheets or LibreOffice Calc. A formula starts with =, uses * for × and / for ÷, and can refer to other cells by their address (like B2, meaning column B, row 2).

Spreadsheets follow the order of operations, so a formula means exactly what the same expression means on paper:

• =2+3*4 gives 14.
• =(2+3)*4 gives 20.

Imagine a kirana bill where column B holds the quantity and column C the price. Row 2 is rice (5 kg at ₹60), row 3 is dal (2 kg at ₹110) and row 4 is oil (1 bottle at ₹180). The total is

=B2C2+B3C3+B4*C4, which is 5 × 60 + 2 × 110 + 1 × 180 = ₹700.

No brackets are needed, because every product is done before the adding, exactly as on paper.

Teachers who mark thousands of answers see the same order-of-operations errors again and again. Knowing them by name is the fastest way to stop making them. Each callout below is a real, common error, with the thinking behind it and the fix.

A class is cooking suji halwa for a school fair. The recipe for one batch needs 1 kg of suji (₹50), 1 kg of sugar (₹45) and 500 g of ghee (₹300). They make 4 batches and sell the halwa in 20 cups. What must each cup cost just to cover the ingredients?

• Cost of one batch: 50 + 45 + 300.
• Cost of 4 batches: 4 × (50 + 45 + 300).
• Cost per cup: 4 × (50 + 45 + 300) ÷ 20.

4 × (50 + 45 + 300) ÷ 20 = 4 × (95 + 300) ÷ 20 = 4 × 395 ÷ 20 = 1,580 ÷ 20 = 79. So each cup must be sold for at least ₹79 to break even.