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§ Arithmetic

Order of Operations

§ Arithmetic

Order of Operations

CCSS.5.OACCSS.6.EE3 min read

Order of operations is a set of rules that determines which mathematical operations to perform first in an expression containing multiple operations. The standard convention follows PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Without these rules, the expression 3 + 4 × 2 could equal either 11 or 14, creating mathematical ambiguity.

§ 01

Why it matters

Order of operations ensures mathematical expressions have a single, unambiguous answer across all contexts. In programming, calculators follow PEMDAS to execute formulas correctly — a payroll system calculating 40 hours × $15 per hour + $50 bonus must compute 40 × 15 + 50 = $650, not 40 × (15 + 50) = $2,600. Scientific formulas rely on precise operation order: calculating kinetic energy as ½mv² requires squaring velocity before multiplying by mass. Financial calculations like compound interest use exponents that must be evaluated before multiplication. Architecture and engineering depend on correct formula evaluation when calculating loads, angles, and measurements. Students encounter order of operations in CCSS 5.OA when evaluating expressions with parentheses, building foundation skills for algebra, where expressions like 3x + 2(x - 4) require systematic evaluation order.

§ 02

How to solve order of operations

Order of Operations (PEMDAS)

  • Parentheses first.
  • Then exponents.
  • Then multiplication and division (left to right).
  • Then addition and subtraction (left to right).

Example: 3 + 4 × 2 = 3 + 8 = 11 (not 14).

§ 03

Worked examples

Beginner§ 01

Which operation do you do first in: 3 + 3 × 2?

Answer: Multiplication: 3 × 2 = 6

  1. Multiply first → 3 × 2 = 6 — Multiplication before addition (PEMDAS).
  2. Then add → 3 + 6 = 9 — Now add the remaining term.
  3. Verify → 3 + 3 × 2 = 9 ✓ — Check the answer.
Easy§ 02

9 packs of 7 stickers, plus 6 extra. How many stickers in total?

Answer: 69

  1. Multiply packs by stickers → 9 × 7 = 63 — Find total stickers in packs first.
  2. Add the extras → 63 + 6 = 69 — Then add the extra stickers.
  3. Verify → 9 × 7 + 6 = 69 ✓ — Check.
Medium§ 03

What is different about 7 × (8 + 3) vs 7 × 8 + 3? Calculate both.

Answer: 7 × (8 + 3) = 77, 7 × 8 + 3 = 59

  1. With parentheses → 7 × (8 + 3) = 7 × 11 = 77 — Parentheses force addition first.
  2. Without parentheses → 7 × 8 + 3 = 56 + 3 = 59 — Without parentheses, multiplication happens first.
  3. Difference → 77 − 59 = 18 — Parentheses change the result.
§ 04

Common mistakes

  • Working left to right without regard for operation priority produces 3 + 4 × 2 = 14 instead of the correct answer 11
  • Ignoring parentheses leads to calculating 7 × 8 + 3 = 59 when the expression 7 × (8 + 3) actually equals 77
  • Treating exponents as regular multiplication gives 2 × 3² = 36 instead of the correct result 2 × 9 = 18
§ 05

Frequently asked questions

What does PEMDAS stand for?
PEMDAS stands for Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Some remember it with the phrase 'Please Excuse My Dear Aunt Sally.' Multiplication and division have equal priority, as do addition and subtraction, so these pairs are evaluated from left to right.
Why do multiplication and division have equal priority?
Multiplication and division are inverse operations with the same priority level. When both appear in an expression like 12 ÷ 3 × 2, they're evaluated left to right: 12 ÷ 3 = 4, then 4 × 2 = 8. This gives 8, not 12 ÷ 6 = 2, which would result from doing multiplication first.
How do you remember the order of operations?
PEMDAS provides the standard memory aid: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. The phrase 'Please Excuse My Dear Aunt Sally' helps recall the sequence. Remember that operations at the same level (like multiplication/division) work left to right, not in strict MD then AS order.
What happens if there are no parentheses in an expression?
Without parentheses, follow the remaining PEMDAS rules: exponents first, then multiplication and division (left to right), finally addition and subtraction (left to right). For example, 5 + 3² × 2 becomes 5 + 9 × 2 = 5 + 18 = 23. The exponent 3² = 9 is calculated first, then multiplication, then addition.
Do calculators automatically follow order of operations?
Most scientific and graphing calculators follow PEMDAS automatically, but basic four-function calculators often calculate left to right regardless of operation priority. Entering 3 + 4 × 2 on a basic calculator might display 14 instead of the mathematically correct answer 11. Always verify calculator behavior with known expressions.
§ 06

See also

SetExpressionMultipleExponentFunction
§ 06

Where to next?

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