What is the correct order of operations for expressions with addition and multiplication?
Executive summary
The agreed convention is: do parentheses (grouping) first, then exponents/orders, then multiplication and division (together, left to right), and finally addition and subtraction (together, left to right); in particular multiplication is performed before addition when both appear without parentheses (e.g., 2 + 3 × 10 = 2 + (3 × 10)) [1] [2]. Several education sites and textbooks teach this using acronyms such as PEMDAS/BEDMAS/GEMDAS and explicitly note that M and D are interchangeable and A and S are interchangeable, processed left-to-right within their tiers [3] [4] [5].
1. Why a rule exists: avoid ambiguity in arithmetic
Mathematicians and educators codified an order of operations so everyone evaluates the same expression the same way; without it expressions like 4 + 2 × 3 could mean different numbers depending on reading order, so the modern algebraic convention gives multiplication higher precedence than addition and encodes grouping with parentheses to force other orders [1] [2].
2. The short, practical rule you’ll see in classrooms
Textbooks and teaching sites present the rule as: Parentheses (or grouping), Exponents (orders), Multiplication and Division (same rank, left-to-right), Addition and Subtraction (same rank, left-to-right). Memory aids vary by country — PEMDAS, BEDMAS, GEMDAS, etc. — but all emphasize that multiplication/division come before addition/subtraction and that you resolve same-level operations left-to-right [6] [5] [3].
3. What “multiplication before addition” really means in examples
When you have 2 + 3 × 10 you must compute 3 × 10 first and then add 2, giving 32; the multiplication has higher precedence and is not overridden by left-to-right reading unless parentheses force a different grouping [2] [1]. Equally, in 3 + 8 × 2 − 6 you do 8 × 2 = 16 first, then 3 + 16 − 6 evaluated left-to-right for addition/subtraction as 19 − 6 = 13 [3].
4. The left-to-right nuance people commonly miss
Multiplication and division are the same precedence; you don’t always multiply before you divide — you perform them in the order they appear from left to right. The same holds for addition and subtraction. For example, 24 ÷ 8 × 2 equals (24 ÷ 8) × 2, not 24 ÷ (8 × 2) [4] [7].
5. Parentheses change everything — and that’s intentional
Brackets or parentheses explicitly change evaluation order: (2 + 3) × 4 forces addition before multiplication; textbooks teach using grouping to avoid ambiguity and to express intended order clearly [1] [8]. When ambiguity would cause different answers, the correct remedy is to add parentheses, not to reinterpret the precedence rules [1].
6. Where conventions differ and why you might see apparent contradictions
Some calculators, programming languages, or historical notations can use different parsing conventions; but standard modern algebraic notation grants multiplication/division higher precedence than addition/subtraction and has used that hierarchy since algebraic notation developed in the 1600s [1]. Education pages also point out different mnemonic names across regions (e.g., BEDMAS in Canada) while maintaining the same operational hierarchy [3] [5].
7. Teaching emphasis and common student mistakes
Education resources emphasize that addition/subtraction and multiplication/division are paired: treat them as two tiers and process ties left-to-right. Students commonly misapply PEMDAS as a strict left-to-right list (do M then D always, or A then S always), which leads to errors; reliable teaching materials stress the left-to-right rule within the M/D and A/S tiers [4] [7].
8. Bottom line and best practice
Follow grouping (parentheses) first, then exponents, then do any multiplications and divisions in the order they appear left-to-right, then do additions and subtractions left-to-right. When in doubt, add parentheses to show the intended order — that eliminates ambiguity entirely [5] [1].
Limitations: available sources do not mention every programming-language exception or specific calculator implementations; check the manual for a given calculator or language if you suspect it treats precedence differently (not found in current reporting).