You buy 2 comics at £3 each and a £4 magazine. In your head that is 2 \times 3 + 4 = £10 — but only if you multiply before you add. Get the order wrong and you would hand over the wrong money. Whenever a calculation mixes different operations, we have to agree on what to do first.

Most sums use a single operation at a time. But what should you do with an expression that mixes them, like 2 + 3 \times 4? Where do you even start?

One person reads left to right and adds first — 2 + 3 = 5, then 5 \times 4 = 20. Another multiplies first — 3 \times 4 = 12, then 2 + 12 = 14. Same sum, but two different answers — 20 and 14. They can't both be right!

If everyone evaluated sums in their own private order, a single expression would mean different things to different people, and maths would stop working. So that everyone — every teacher, every textbook, every calculator on Earth — gets the same answer, mathematicians long ago agreed on one fixed order of operations for the four operations: addition (+), subtraction (-), multiplication (\times), and division (\div). It isn't that one order is "more true" than another — it's a shared agreement, like everyone driving on the same side of the road.

Think of a bus queue. It only works because everybody agrees who goes first — if half the people invented their own rule there'd be chaos. The order of operations is maths' queue: a single agreed line-up so that 2 + 3 \times 4 means exactly one thing, no matter who reads it. The rule below is that queue, written down.

Strong operations go first

The rule sorts the four operations into two tiers. Multiplication and division are the "strong" pair — they happen first. Addition and subtraction are the "weak" pair and come after.

So 2 + 3 \times 4 = 14, not 20 — the multiplication is done before the addition, even though the addition is written first. Press play to watch it unfold (a fresh expression each time):

See it: the order in action

Here is the same idea you can step through yourself. The expression appears on top; the diagram circles the operation that goes first, works it out, and only then finishes with the weaker operation. Press Refresh for a brand-new expression.

A few worked examples

For each one, compare the wrong left-to-right rush with the right order:

• 2 + 3 \times 4: wrong is 5 \times 4 = 20; right is 2 + 12 = \mathbf{14} (multiply first).
• 20 - 4 \times 3: wrong is 16 \times 3 = 48; right is 20 - 12 = \mathbf{8} (multiply first, then subtract).
• 18 \div 3 + 1: wrong is 18 \div 4; right is 6 + 1 = \mathbf{7} (divide first).

Same tier? Left to right

When the operations are on the same tier, neither is stronger — so you just work left to right, like reading a sentence.

(Not 10 - 7 = 3 — the subtraction is further left, so it goes first.) Multiplication and division share their tier the same way:

(Not 12 \div 6 = 2 — the division is on the left, so it goes first.)

Brackets jump the queue

What if you really do want to add first? Wrap it in brackets (also called parentheses, the round ones). Whatever sits inside brackets is worked out before anything else — they let you push a weak operation to the front of the queue:

Same numbers, same operations — but the brackets flip the answer from 14 to 20. That is exactly what they are for.

The whole rule, in order

1. Brackets — innermost first.
2. Orders / Indices — powers like 3^2 (a lesson for later).
3. Multiplication and division — then these, left to right.
4. Addition and subtraction — last of all, left to right.

A handy way to remember it is the word BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction.

Different countries teach the very same rule with different words. In the UK you'll hear BODMAS or BIDMAS — the O ("Orders") and the I ("Indices") both just mean powers. In the USA it's PEMDAS ("Please Excuse My Dear Aunt Sally"), where P is Parentheses and E is Exponents. The letters move around, but the queue is identical everywhere on the planet: brackets, then powers, then \times \div, then + -.

"Orders" (or "Indices", or "Exponents") are powers — a tiny number that tells you how many times to multiply something by itself, like 3^2 = 3 \times 3 = 9. They blast a number upward fast, so they sit near the front of the queue: right after brackets and before ordinary multiplying. You don't need them yet — just know that's the slot the O / I / E fills in the mnemonics.

See it explained