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Order of Operations

Grades 5–8 Arithmetic rules

2 + 3 × 4 — is it 20 or 14? Math needs a shared convention so everyone reads the same expression the same way. That convention is the order of operations: multiplication before addition, so the answer is 14. It's not a law of nature — it's punctuation, agreed on so writing math stays unambiguous.

What PEMDAS really means

P parentheses first E exponents MD × and ÷ together — left to right, equal rank AS + and − together — left to right, equal rank
The part everyone forgets Multiplication does not come before division, and addition does not come before subtraction. They're equal-rank pairs done left to right:
8 ÷ 4 × 2 = 2 × 2 = 4 (not 8 ÷ 8 = 1) 10 − 4 + 2 = 6 + 2 = 8 (not 10 − 6 = 4)

Worked example

3 + 4 × (2 + 5)² − 8 parentheses: (2 + 5) = 7 exponent: 7² = 49 multiply: 4 × 49 = 196 left to right: 3 + 196 − 8 = 191

The famous ambiguous one

You've seen it online: 8 ÷ 2(2+2). Strict left-to-right gives 16; treating 2(4) as a glued unit gives 1. Real mathematicians avoid the ÷ symbol precisely because it causes this — a horizontal fraction bar removes all doubt. When in doubt, write parentheses. The rule was never meant to settle trick questions; it exists to keep honest writing clear.

Why the convention exists at all Notation evolves to fit the structure of the math. Exponentiation is repeated multiplication, multiplication is repeated addition — so the "strongest" operation binds tightest, and expressions read naturally. For perspective on how notation and convention shape what counts as good mathematics:
  • Tao, T. (2007). What is good mathematics? Bulletin of the American Mathematical Society, 44(4), 623–634. doi:10.1090/S0273-0979-07-01168-8 · arXiv:math/0702396 · MathSciNet: MR2338369
  • Cajori, F. (1928–29). A History of Mathematical Notations. Open Court — the standard history of how symbols like ÷, ×, and parentheses were chosen.
Try it yourself:
  1. 18 − 3 × 4  (answer: 6)
  2. 24 ÷ 6 × 2  (answer: 8 — left to right)
  3. 2 + 3² × 2  (answer: 20)

Related: negative numbers · word problems where operation order is the whole game.

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