How do you apply PEMDAS to expressions with multiplication and addition?
Executive summary
PEMDAS is a shorthand to resolve ambiguity in multi-operation expressions: do Parentheses, then Exponents, then Multiplication/Division (left-to-right), then Addition/Subtraction (left-to-right) — which means multiplication does not “trump” addition by whim but by formal precedence; parentheses can override that order [1] [2]. The common classroom focus on the letters can mislead students about equal priorities (M and D are equal; A and S are equal) and the critical left-to-right rule when operations share rank [1] [3].
1. What PEMDAS actually instructs and why multiplication sits above addition
PEMDAS stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction and encodes a historical convention that keeps algebra unambiguous and compact: multiplication is evaluated before addition because the distributive law of algebra gives multiplication a natural higher precedence in notation [4], so an expression like 1 + 2 × 3 must be read as 1 + (2 × 3) = 7 rather than (1 + 2) × 3 = 9 [4].
2. Multiplication and division: a tied ranking resolved left-to-right
Although PEMDAS lists M before D, multiplication and division have the same priority and are performed in the order they appear from left to right; the same applies to addition and subtraction later on [1] [2]. For example, 12 ÷ 6 × 3 should be done as (12 ÷ 6) × 3 = 6, not 12 ÷ (6 × 3) = 0.666…, illustrating that following left-to-right for equal-ranked operations is essential [1] [5].
3. Addition comes after multiplication, but parentheses can flip the script
By default, multiply before you add, so 2 × 4 + 7 = (2 × 4) + 7 = 15 [6]. However, parentheses change precedence: (2 + 4) × 7 forces the addition first and yields 42, demonstrating that grouping symbols are the explicit mechanism for overriding PEMDAS conventions when a different order is intended [7] [4].
4. Common pitfalls, misleading mnemonics and how to avoid them
Mnemonics like “Please Excuse My Dear Aunt Sally” help memory but can mislead: students sometimes think M must always be done before D or A always before S, which is wrong; the correct nuance is that M and D share priority and so do A and S, each pair handled left-to-right [3] [2]. Another trap is reading strictly left-to-right for all operations; that only works if operations share the same precedence — otherwise it produces errors, e.g., 2 + 5 × 3 is 17, not 21 [2] [6].
5. Practical takeaways and a small decision rule for everyday problems
When faced with an expression: first evaluate anything inside parentheses or other grouping symbols, then compute exponents, then scan left-to-right performing any multiplication or division as encountered, then scan left-to-right performing any addition or subtraction — if clarity is desired, add parentheses to force the intended order [1] [8]. Calculators and software follow the same precedence conventions, though nuance exists for exponent associativity in some tools (a^b^c may be grouped differently across platforms), so when results disagree, check grouping explicitly [4].
6. Alternative viewpoints, historical context and why teaching nuance matters
Historical sources show that the precedence rules evolved over centuries and that disagreements persisted into the early 20th century, which explains why different countries use different acronyms (PEMDAS, BODMAS, BEDMAS) and why educators emphasize the left-to-right tie-breaking rule to prevent misapplication of mnemonics [4] [9]. Present-day math education guidance stresses understanding the underlying rules — not just memorizing letters — because that understanding prevents mistakes in algebra, programming contexts, and cross-platform computations [4] [8].