Order of Operations

Updated 4 October 2026

Here is the argument that breaks the internet every few months: 2 + 3 × 4. Half the comments say 20, half say 14, and the half that says 14 is right — not because a committee voted, but because 3 × 4 was never two numbers. It was a package.

What is the order of operations?

The order of operations is the rule for which part of a calculation comes first: brackets, then powers and roots, then multiplication and division from left to right, then addition and subtraction from left to right. In 20 − 2 × (3 + 4), the brackets come first and give 7, the multiplication gives 2 × 7 = 14, and the subtraction gives 20 − 14 = 6. So 20 − 2 × (3 + 4) = 6. The rule exists so everyone reads a calculation the same way. PEMDAS, BODMAS and BIDMAS are this one list under different names. People often slip on a tier with two operations in it. Multiplication and division are equal partners, so work them from left to right: 12 ÷ 3 × 2 = 4 × 2 = 8, not 2. To check, write brackets around each step in the order you did it: 20 − (2 × (3 + 4)) = 6.

1. Brackets first. Anything in brackets is worked out before it can touch the rest.
2. Then × and ÷, left to right. One tier, one sweep. (Powers like $3^2$3² sit just above this tier — square first, then multiply.)
3. Then + and −, left to right. Also one tier, one sweep.

That is the whole rule.

Why the order exists

Brackets exist for the opposite moment: when you want the addition to happen first, you say so out loud. (2 + 3) × 4 = 5 × 4 = 20. Same three numbers, different sentence.

How does the order of operations work on an example?

Work an example one tier at a time. In 4 + 6 × 2 − 3, the multiplication goes first: 6 × 2 = 12, so the line becomes 4 + 12 − 3. Then add and subtract from left to right: 4 + 12 = 16, and 16 − 3 = 13. So 4 + 6 × 2 − 3 = 13. Working straight from left to right would reach 17, which is wrong, because 6 × 2 is one amount that must be found before anything is added to it. Brackets are the case that changes the order: (4 + 6) × 2 − 3 tells you to add first, so it becomes 10 × 2 − 3 = 20 − 3 = 17. A power works like a bracket one tier higher: in $2+3^2$2 + 3², square first, so 2 + 9 = 11. To check, write the brackets in: 4 + (6 × 2) − 3 = 13.

Step 2 beats step 3: multiply before you add

$2+3\times 4\to 2+12\to$2 + 3 × 4 → 2 + 12 → 14

$5\times 2+3\times 4\to 10+12\to$5 × 2 + 3 × 4 → 10 + 12 → 22 — both packages open first, then the adding.

Brackets change the sentence

$\left(2+3\right)\times 4\to 5\times 4\to$(2 + 3) × 4 → 5 × 4 → 20

$18-\left(4+6\right)\to 18-10\to$18 − (4 + 6) → 18 − 10 → 8

All four operations at once

Work the tiers in order: brackets, then × and ÷, then + and −.

$7+2\times 6-5\to 7+12-5\to 19-5\to$7 + 2 × 6 − 5 → 7 + 12 − 5 → 19 − 5 → 14

With a power

$3^2+4\times 5\to 9+20\to$3² + 4 × 5 → 9 + 20 → 29 — the square unwraps first (it is compressed multiplication, one level further up), then the ×, then the +.

What do people get wrong about the order of operations?

The most common mistake is reading PEMDAS as six separate steps, as if all multiplication came before all division and all addition before all subtraction. In fact multiplication and division share one tier, addition and subtraction share another, and each tier is worked from left to right. So 12 ÷ 2 × 3 = 6 × 3 = 18, while doing the multiplication first would give 12 ÷ 6 = 2, which is wrong. Likewise 10 − 4 + 3 = 6 + 3 = 9, while adding first gives 10 − 7 = 3. The pairs share a tier because division undoes multiplication and subtraction undoes addition, so neither one outranks its partner. The second mistake is working right to left inside a tier, which only changes the answer when division or subtraction is involved. To check a line with subtraction in it, rewrite each subtraction as adding a negative, and then the order no longer matters: 10 + (−4) + 3 = 9.

• Reading the letters as six steps. The M and D in PEMDAS are one tier, left to right — and so are A and S. 8 ÷ 2 × 4 = 4 × 4 = 16, not 1. 8 − 2 + 5 = 6 + 5 = 11, not 1.
• Sweeping right to left. Within a tier the sweep is left to right.