How BIDMAS sets the order of operations, step by step

BIDMAS gives the order of operations: brackets first, then indices, then division and multiplication, then addition and subtraction. Follow it and 2 + 1 × 3 is 5, not 9, because the multiplication comes before the addition.

The letters are easy to learn. The mistakes come from two details: division and multiplication share the same rank, and so do addition and subtraction, so each pair is worked from left to right. This guide explains each letter, the left-to-right rule, how brackets and powers change a calculation, and where the order of operations appears in the national curriculum and at GCSE, with questions to practise at the end.

What BIDMAS stands for

Each letter is a step, taken in this order:

The national curriculum uses exactly this kind of example. In the guidance for Year 6, the mathematics programmes of study show that 2 + 1 × 3 = 5 while (2 + 1) × 3 = 9, and the statutory requirement asks pupils to use their knowledge of the order of operations to carry out calculations involving the four operations.

The reason for a fixed order is simple: without one, the same line of symbols could give two answers. With BIDMAS, everyone who reads 20 − 3 × 4 gets 8, never 68.

BIDMAS rules: the left-to-right trap

The most common error is to read BIDMAS as six separate steps and do every multiplication before any division. Division and multiplication have the same rank, so you work through them from left to right, in the order they appear.

Take 12 ÷ 3 × 2. Left to right, 12 ÷ 3 is 4, and 4 × 2 is 8. Doing the multiplication first would give 12 ÷ 6 = 2, which is wrong. The same goes for addition and subtraction: 10 − 4 + 3 is 9, because 10 − 4 comes first. Adding first would give 10 − 7 = 3.

A longer example shows the whole method. In 100 − 60 ÷ 3 × 2, there are no brackets and no powers, so start with division and multiplication, left to right: 60 ÷ 3 = 20, then 20 × 2 = 40. Only then subtract: 100 − 40 = 60. Writing each step on a new line keeps the order visible and makes errors easy to spot.

Spreadsheets use the same ranks. Typed into a cell as =10-4+3, the formula returns 9, and =12/3*2 returns 8, because the program also works from left to right within a rank. If a spreadsheet gives an answer you did not expect, the missing brackets are usually the reason.

BODMAS or BIDMAS, and why they agree

Some schools teach BODMAS instead of BIDMAS. The O stands for orders, another word for powers and roots, so the two names describe the same rules. In the United States, the same order is often taught as PEMDAS, with parentheses for brackets and exponents for indices.

All three versions share the same weakness if they are learned as a list: they put D before M, or M before D, and A before S. The letters give the ranks, not the order inside a rank. A safer way to remember it is as four levels: brackets, then powers, then multiplication and division together, then addition and subtraction together.

The order matters most with mental arithmetic, because there is no calculator to apply it for you. In 6 + 18 ÷ 3², the power comes first, 3² = 9, then the division, 18 ÷ 9 = 2, and the addition last: 8.

Order of operations with brackets and powers

Brackets change the order on purpose. 3 + 2² × 5 is 3 + 4 × 5 = 23, but (3 + 2)² × 5 is 5² × 5 = 125. When brackets sit inside other brackets, start with the innermost pair and work outwards.

Two situations work like hidden brackets:

• A fraction bar. In (8 + 4) over (2 × 3), the top and the bottom are each worked out first: 12 ÷ 6 = 2. Written on one line, it needs brackets around both parts.
• A negative number and a power. −3² means −(3²), so it equals −9. To square −3, write (−3)², which equals 9. This detail costs marks in algebra, when a negative value is substituted into a formula.

Powers also come before multiplication: 4 × (5 − 2)² is 4 × 3² = 4 × 9 = 36. If you multiplied first, you would square 12 instead and get 144.

Where BIDMAS appears in school and at GCSE

The order of operations starts at primary school, with the Year 6 requirement quoted above, and continues in secondary school. The key stage 3 programme of study asks pupils to use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals. Our guide to maths in Year 7 explains what else that programme covers.

At GCSE, the AQA specification for GCSE Mathematics (8300) lists the same skill, priority of operations with brackets, powers, roots and reciprocals, in its basic foundation content, so every student is expected to master it. It matters most on the paper without a calculator, described in our article on the GCSE non-calculator paper.

A scientific calculator applies the order of operations when you type a whole expression, but a basic calculator may simply work through the keys in the order you press them. Knowing the rules lets you check either one.

BIDMAS questions to practise

Cover the answers, work each question on paper or in your head, then check the first step if you went wrong.

Most of these questions also test multiplication facts. If 7 × 8 or 9 × 6 still takes a moment, the multiplication tables from 2 to 12 are the best place to start. Kalc trains times tables, fractions, percentages and algebra in about ten minutes a day, and short daily practice is what makes the order of operations automatic.

When you practise, say the rank out loud before each step: brackets, powers, multiply and divide, add and subtract. After a few days the habit replaces the list, and long expressions stop being intimidating.

Frequently asked questions

What does BIDMAS stand for?

Brackets, indices, division, multiplication, addition and subtraction. Division and multiplication share the same rank, as do addition and subtraction, and each pair is worked from left to right.

Do you divide or multiply first in BIDMAS?

Neither comes first: division and multiplication have the same rank, so you work them from left to right. 12 ÷ 3 × 2 is 4 × 2, which equals 8.

Is BODMAS the same as BIDMAS?

Yes. In BODMAS the O stands for orders, which means powers and roots, the same as indices in BIDMAS. The rules are identical.

What is −3² according to the order of operations?

−3² equals −9, because the power applies to 3 before the minus sign. To square negative three, write (−3)², which equals 9.

Where is the order of operations taught in the national curriculum?

In Year 6, pupils use the order of operations with the four operations. At key stage 3 and GCSE, it extends to brackets, powers, roots and reciprocals.