Remember that sinking feeling in math class when you got a problem wrong because "it looked right"? I do. Back in 8th grade, I failed a quiz spectacularly because I calculated 10 - 2 × 3 as 24 instead of 4. My teacher circled it in angry red pen: "ORDER OF OPERATIONS!" That moment burned the math order of operations into my brain forever.
Let's fix that confusion once and for all. The math order of operations isn't some arbitrary rule mathematicians invented to torture students. It's the GPS for math expressions – without it, we'd have mathematical chaos where 2 + 3 × 4 could equal 20 or 14 depending on who's calculating. Not ideal when building bridges or calculating medicine doses!
Why the Math Order of Operations Actually Matters
Think about reading English without punctuation. "Let's eat grandma" vs "Let's eat, grandma" – big difference! The math order of operations is punctuation for numbers. It creates universal understanding.
Before PEMDAS became standard in the early 20th century, math was like the Wild West. A 1912 algebra textbook might tell you to solve from left to right, while another said multiplication always comes first. Can you imagine engineers trying to collaborate like that? Scary stuff.
Here's what happens without consistent rules:
8 ÷ 2(2 + 2) = ?
• Left-to-right: 8 ÷ 2 = 4, then 4 × 4 = 16
• Multiplication first: 2 × 4 = 8, then 8 ÷ 8 = 1
Both seem logical! This viral math problem broke the internet because people forgot the math order of operations rules.
• Left-to-right: 8 ÷ 2 = 4, then 4 × 4 = 16
• Multiplication first: 2 × 4 = 8, then 8 ÷ 8 = 1
Both seem logical! This viral math problem broke the internet because people forgot the math order of operations rules.
The Golden Rule: PEMDAS/BODMAS Demystified
You've probably heard the acronyms PEMDAS (US) or BODMAS (UK). They're two sides of the same coin:
| PEMDAS | BODMAS | Meaning | Real-World Example |
|---|---|---|---|
| Parentheses | Brackets | Grouping symbols first | Tax calculations: (base salary + bonus) × tax rate |
| Exponents | Orders | Powers & roots | Area formulas: πr² where r=5 |
| Multiplication Division |
Division Multiplication |
Left to right (equal priority) | Shopping: 3 shirts × $20 + 2 pants ÷ 2 (50% off) |
| Addition Subtraction |
Addition Subtraction |
Left to right (equal priority) | Budgeting: Income - rent + side hustle money |
Critical insight most teachers skip: Multiplication/division have equal priority, as do addition/subtraction. You solve whichever comes first left to right. PEMDAS makes it look like multiplication always comes before division, but that's misleading.
⚠️ My Biggest Pet Peeve: Some textbooks present PEMDAS as strict sequential steps. This causes so many errors! Remember:
6 ÷ 2 × 3 = 9 (not 1!) because division and multiplication share priority.
6 ÷ 2 × 3 = 9 (not 1!) because division and multiplication share priority.
Where Humans Get Tripped Up (Including Me!)
After tutoring for 15 years, I see these mistakes repeatedly:
- Implied multiplication: Does 8 ÷ 2x mean 8 ÷ (2x) or (8 ÷ 2)x? Technically the latter, but scientists often use the former. My rule: when in doubt, add parentheses!
- Fractions bars as grouping symbols: In \(\frac{6}{2(1+2)}\), the denominator is entirely below the bar. But written as 6/2(1+2)? Chaos ensues.
- Exponents with parentheses: (-3)² = 9 vs -3² = -9. That negative sign causes endless grief.
Walkthrough: Solving Problems Step-by-Step
Let's apply the math order of operations to real expressions. I'll show my scratch work like I'm teaching a study buddy:
Example 1: Basic Application
Expression: 12 ÷ 3 × 2 + 5² - (6 - 4)
My thought process:
"Okay, parentheses first: (6 - 4) = 2
Exponents next: 5 squared is 25
Now multiplication/division left to right: 12 ÷ 3 = 4, then 4 × 2 = 8
Finally addition/subtraction: 8 + 25 = 33, then 33 - 2 = 31"
Solution: 31
My thought process:
"Okay, parentheses first: (6 - 4) = 2
Exponents next: 5 squared is 25
Now multiplication/division left to right: 12 ÷ 3 = 4, then 4 × 2 = 8
Finally addition/subtraction: 8 + 25 = 33, then 33 - 2 = 31"
Solution: 31
Example 2: The Viral Stumper
Expression: 20 ÷ 5(3 + 1)
Why people fight:
• Team PEMDAS: "Parentheses first: 3+1=4. Then multiplication: 5×4=20. Then division: 20÷20=1"
• Team Left-to-Right: "20÷5=4, then 4×4=16"
Technical answer: According to modern rules, implied multiplication doesn't have priority. So 20 ÷ 5 × 4 = 16.
My advice: Avoid ambiguity! Write either (20÷5)×(3+1) or 20÷[5(3+1)]
Why people fight:
• Team PEMDAS: "Parentheses first: 3+1=4. Then multiplication: 5×4=20. Then division: 20÷20=1"
• Team Left-to-Right: "20÷5=4, then 4×4=16"
Technical answer: According to modern rules, implied multiplication doesn't have priority. So 20 ÷ 5 × 4 = 16.
My advice: Avoid ambiguity! Write either (20÷5)×(3+1) or 20÷[5(3+1)]
Example 3: Real-World Math
Calculating pizza cost per person:
2 pizzas × $15 each + $5 delivery fee ÷ 4 people
Wrong: 2×15=30, 30+5=35, 35÷4=$8.75
Right: Delivery fee is added last! (2×15 + 5) ÷ 4 = (30+5)÷4 = $8.75
Surprise! Same answer here, but only by luck. Change the numbers: 3 pizzas × $12 + $10 tip ÷ 6 people
• Without parentheses: 3×12=36, 36+10=46, 46÷6≈$7.67
• Actual cost per person: (36 + 10)÷6 ≈ $7.67? Wait no! (3×12 + 10)÷6 = (36+10)÷6 = 46÷6 ≈ $7.67
• But if you meant tip divided among all: 3×12 + (10÷6) ≈ 36 + 1.67 = $37.67 total!
Moral: Group intentions matter more than rules!
Wrong: 2×15=30, 30+5=35, 35÷4=$8.75
Right: Delivery fee is added last! (2×15 + 5) ÷ 4 = (30+5)÷4 = $8.75
Surprise! Same answer here, but only by luck. Change the numbers: 3 pizzas × $12 + $10 tip ÷ 6 people
• Without parentheses: 3×12=36, 36+10=46, 46÷6≈$7.67
• Actual cost per person: (36 + 10)÷6 ≈ $7.67? Wait no! (3×12 + 10)÷6 = (36+10)÷6 = 46÷6 ≈ $7.67
• But if you meant tip divided among all: 3×12 + (10÷6) ≈ 36 + 1.67 = $37.67 total!
Moral: Group intentions matter more than rules!
Advanced Scenarios & Pro Tips
When you encounter these, slow down:
| Scenario | Proper Approach | Common Mistake |
|---|---|---|
| Nested parentheses | Work from innermost outward: [2 + {3×(4-1)}] | Solving outer brackets first |
| Fraction bars | Treat numerator and denominator as grouped | Forgetting the "invisible parentheses" |
| Radicals (√) | √9×4 = (√9)×4=3×4=12, not √36=6! | Mistaking √ as grouping symbol |
| Absolute values | | | Solve inside before applying absolute | Ignoring internal operations |
Pro trick I use: Rewrite expressions with excessive parentheses first, then remove redundant ones. For example:
Original: 3 + 4 × 2² ÷ (1 + 1)
Safe rewrite: 3 + [ (4 × (2²)) ÷ (1 + 1) ]
Calculate: 3 + [ (4 × 4) ÷ 2 ] = 3 + [16 ÷ 2] = 3 + 8 = 11
Safe rewrite: 3 + [ (4 × (2²)) ÷ (1 + 1) ]
Calculate: 3 + [ (4 × 4) ÷ 2 ] = 3 + [16 ÷ 2] = 3 + 8 = 11
Why Your Calculator Betrays You
Ever type 3 + 4 × 2 into different calculators? Some give 14, others 11! Here's why:
- Basic calculators: Often operate strictly left-to-right (3+4=7, 7×2=14)
- Scientific calculators: Follow proper math order of operations (4×2=8, 3+8=11)
- Phone calculators: Most modern ones use PEMDAS, but test yours!
I learned this the hard way balancing my checkbook in college. My $5 calculator told me I had $50 more than reality. Ramen noodles for a week thanks to bad math order!
FAQs: Your Burning Questions Answered
Does multiplication always come before division in PEMDAS?
Absolutely not! This misconception causes 60% of errors I see. PEMDAS groups M/D together. In 8 ÷ 4 × 2, you divide first (8÷4=2) then multiply (2×2=4). If multiplication came first, you'd get 8 ÷ 8 = 1 (wrong!).
How do roots fit into the math order of operations?
Roots fall under "exponents" (the E in PEMDAS). √36 + 4 means (√36) + 4 = 6 + 4 = 10. Not √40! The root applies ONLY to the number under its symbol unless grouping is used.
What about functions like sin/cos/log?
Functions apply to what's immediately after them. In sin30° × 2, calculate sin30° first (0.5), then multiply. But in sin(30° × 2), multiply first: sin60° ≈ 0.866. Placement matters!
Why do Americans say PEMDAS and Brits say BODMAS?
Just linguistic differences! BODMAS means Brackets, Orders, Division/Multiplication, Addition/Subtraction. Orders = exponents/roots. The rules are identical despite the acronym.
How do I handle expressions with multiple exponents?
Work from top down: 2^3^2 means 2^(3^2) = 2^9 = 512, not (2^3)^2=64. This "right-associativity" trips up even advanced students!
Teaching Kids Without the Headache
After helping my niece struggle through this, I developed a better approach than rote memorization:
- Use real contexts: "If 4 pizzas cost $40 total, how much per pizza? Now add 2 drinks at $3 each – total cost?" Builds intuition for grouping.
- Color-code expressions: Highlight operations in different colors based on priority.
- Embrace mistakes: Have students deliberately solve incorrectly to see why rules matter. "What if banks used left-to-right math for mortgages?"
Personal confession: I still write extra parentheses in important calculations. My tax spreadsheet looks like it's covered in parentheses confetti. Better safe than sorry!
Beyond Basics: When PEMDAS Isn't Enough
In higher math, conventions get fuzzy:
- Matrices & vectors: Have specialized rules overriding PEMDAS
- Programming languages: Each has its own operator hierarchy (C++ vs Python)
- Advanced notations: In some physics texts, implied multiplication has priority
But for 99% of users? Mastering standard math order of operations covers algebra, finances, recipes, DIY projects – basically everything practical. No need to overcomplicate it!
Final Thoughts: Why This Still Matters
In our calculator-filled world, why learn this? Three reasons:
- Error detection: Spot when your calculator/app gives nonsense results
- Clear communication: Write expressions others interpret correctly
- Logical foundation: Algebra and higher math build directly on this
The math order of operations isn't about rigid rules – it's about shared understanding. Like driving on the right (or left!) side of the road. Sure, you could drive anywhere, but we agree on standards to avoid collisions. Now go solve something confidently!
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