Order of Operations
Order of operations is a set of rules that determines the sequence in which mathematical operations should be performed within an expression. The acronym BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) provides the standard hierarchy that ensures calculations produce consistent results. Without these rules, the expression 3 + 4 × 2 could equal either 11 or 14, creating mathematical ambiguity.
Why it matters
Order of operations appears throughout real-world calculations, from working out shopping bills to engineering formulas. When calculating the cost of 5 items at £8 each plus a £3 delivery charge, the correct sequence (5 × 8 + 3 = £43) differs significantly from the incorrect order (5 × 11 = £55). This foundation becomes crucial in Year 7 algebra, where expressions like 2x + 3y require proper sequencing. GCSE mathematics extensively tests these principles in contexts ranging from compound interest calculations to physics equations. Professional fields including accounting, construction, and computer programming rely on consistent order of operations to ensure accurate results. The rules prevent costly errors in everything from recipe scaling (doubling 2 cups flour plus 1 cup sugar correctly yields 5 cups, not 6) to calculating mortgage payments where multiple operations determine monthly costs.
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).
Worked examples
True or false: 2 + 2 × 3 = 12
Answer: False (8)
- Multiply first → 2 × 3 = 6 — Multiplication before addition (PEMDAS).
- Then add → 2 + 6 = 8 — Now add the remaining term.
- Verify → 2 + 2 × 3 = 8 ✓ — Check the answer.
10 packs of 6 stickers, plus 8 extra. How many stickers in total?
Answer: 68
- Multiply packs by stickers → 10 × 6 = 60 — Find total stickers in packs first.
- Add the extras → 60 + 8 = 68 — Then add the extra stickers.
- Verify → 10 × 6 + 8 = 68 ✓ — Check.
What is different about 10 × (2 + 6) vs 10 × 2 + 6? Calculate both.
Answer: 10 × (2 + 6) = 80, 10 × 2 + 6 = 26
- With parentheses → 10 × (2 + 6) = 10 × 8 = 80 — Parentheses force addition first.
- Without parentheses → 10 × 2 + 6 = 20 + 6 = 26 — Without parentheses, multiplication happens first.
- Difference → 80 − 26 = 54 — Parentheses change the result.
Common mistakes
- Working left to right without considering operation priority leads to errors like calculating 2 + 3 × 4 as 20 instead of 14
- Ignoring brackets changes results dramatically, such as computing 6 × (2 + 3) as 6 × 2 + 3 = 15 instead of 6 × 5 = 30
- Treating division and multiplication as having different priorities causes mistakes like evaluating 12 ÷ 3 × 2 as 12 ÷ 6 = 2 instead of 4 × 2 = 8