Why Calculators Sometimes Give Different Answers: Understanding Order of Operations

Type 6 + 2 × 3 into two different calculators and you might get two different answers: 24, or 12. Only one is mathematically correct — and understanding why reveals one of the most important, and most commonly misapplied, rules in basic math.

The rule: PEMDAS / BODMAS

Mathematics follows a fixed order when evaluating expressions with multiple operations, commonly remembered by the acronym PEMDAS (or BODMAS in some countries):

1. Parentheses / Brackets — solve anything inside brackets first
2. Exponents / Orders — powers and roots
3. Multiplication and Division — left to right, same priority
4. Addition and Subtraction — left to right, same priority

Applying this to 6 + 2 × 3: multiplication happens before addition, so it's 6 + (2 × 3) = 6 + 6 = 12. A calculator that returns 24 is evaluating left to right without respecting operation priority — this is the difference between a "basic" arithmetic calculator and a properly order-aware one.

Why multiplication/division and addition/subtraction are "tied"

A common misconception is that multiplication always comes before division, and addition always before subtraction — but they're actually equal priority pairs, resolved left to right. 20 ÷ 4 × 2 is not 20 ÷ 8; it's (20 ÷ 4) × 2 = 5 × 2 = 10, because division appears first reading left to right.

Where this trips people up in real life

• Spreadsheets: Excel and Google Sheets follow standard order of operations, so a formula like =6+2*3 returns 12, not 24.
• Nested parentheses: (4 + 2) × (10 − 3) requires solving each set of parentheses independently first (6 × 7 = 42) before combining.
• Negative exponents and roots: -2² is often miscalculated. Strictly by order of operations, the exponent applies before the negative sign, so this equals −4, not 4 — because the expression is read as −(2²), not (−2)². Writing (−2)² explicitly changes the answer to 4.

A worked example combining several rules

3 + 4 × (2 + 3)² ÷ 5

1. Parentheses first: (2 + 3) = 5, so: 3 + 4 × 5² ÷ 5
2. Exponent next: 5² = 25, so: 3 + 4 × 25 ÷ 5
3. Multiplication and division, left to right: 4 × 25 = 100, then 100 ÷ 5 = 20, so: 3 + 20
4. Addition last: 3 + 20 = 23

Skipping the order and solving left to right would give a completely different, incorrect answer.

Why this matters beyond school

Order of operations isn't just an academic exercise — it's the foundation every spreadsheet formula, financial model, and piece of software relies on to calculate correctly. A single misapplied operation order in a budget spreadsheet or engineering calculation can produce results that look plausible but are simply wrong. Understanding the rule is the difference between trusting a number because a calculator produced it, and actually knowing why that number is correct.