Order of Operations (PEMDAS)
Without a consistent order, 2 + 3 × 4 could be 20 or 14. PEMDAS ensures everyone gets the same answer.
Why Does Order Matter?
Without rules, 2 + 3 × 4 could mean:
- (2 + 3) × 4 = 5 × 4 = 20 ❌
- 2 + (3 × 4) = 2 + 12 = 14 ✅
PEMDAS (or BODMAS in some countries) is the universal agreement on the order.
PEMDAS Explained
P — Parentheses (and brackets): Do these FIRST
E — Exponents (powers, roots): Do these SECOND
M/D — Multiplication and Division: Equal priority, left to right
A/S — Addition and Subtraction: Equal priority, left to right
Step-by-Step Examples
Example 1: 3 + 4 × 2
Step 1: No parentheses. No exponents.
Step 2: Multiplication: 4 × 2 = 8
Step 3: Addition: 3 + 8 = 11
Example 2: (5 + 3) × 2 - 4
Step 1: Parentheses: 5 + 3 = 8
Step 2: Multiplication: 8 × 2 = 16
Step 3: Subtraction: 16 - 4 = 12
Example 3: 2³ + 5 × (4 - 1)
Step 1: Parentheses: 4 - 1 = 3
Step 2: Exponents: 2³ = 8
Step 3: Multiplication: 5 × 3 = 15
Step 4: Addition: 8 + 15 = 23
Example 4: 24 ÷ 6 × 2
M and D have EQUAL priority — just go left to right:
24 ÷ 6 = 4. Then 4 × 2 = 8
(NOT 24 ÷ 12 = 2 — that would mean you did multiplication first!)
Example 5: 18 - 3 + 7
A and S have EQUAL priority — left to right:
18 - 3 = 15. Then 15 + 7 = 22
Nested Parentheses
Work from the innermost parentheses outward.
Example 6: 2 × [(4 + 3) × 2 - 5]
Step 1: Inner parentheses: 4 + 3 = 7
Step 2: Brackets: 7 × 2 - 5 = 14 - 5 = 9
Step 3: Multiply: 2 × 9 = 18
Implicit Multiplication
When a number is next to parentheses, it means multiply: 5(3) = 5 × 3 = 15
This is still multiplication — same priority as regular ×.
The Famous Internet Debate: 8 ÷ 2(2+2)
Following strict PEMDAS:
Parentheses: 2 + 2 = 4
Left to right: 8 ÷ 2 = 4, then 4 × 4 = 16
Some argue it's 8 ÷ (2×4) = 1, but standard mathematical convention says 16. The takeaway? When in doubt, use more parentheses to make your intention clear!
Memory Tricks
- Please Excuse My Dear Aunt Sally (PEMDAS)
- Brackets, Orders, Division, Multiplication, Addition, Subtraction (BODMAS — used in UK/Australia)
- Think of it as a 4-tier system: Grouping → Powers → Multiply/Divide → Add/Subtract
🎯 Try It Yourself
Test your understanding with these practice problems.
1. Solve: 8 + 2 × 5
💡 Hint: Multiply first: 2×5=10, then add 8
2. Solve: (6 + 4) × 3 - 2
💡 Hint: Parentheses first: 10 × 3 - 2 = 30 - 2
3. Solve: 12 ÷ 4 + 3²
💡 Hint: Exponent: 3²=9. Division: 12÷4=3. Add: 3+9
4. Solve: 2 × (8 - 3)² ÷ 5
💡 Hint: Parentheses: 5. Exponent: 25. Multiply: 50. Divide: 10
5. Solve: 100 - 4 × 3² + 6
💡 Hint: Exponent: 9. Multiply: 36. Then 100 - 36 + 6
⚠️ Common Mistakes
- ✗Working left to right without following PEMDAS. 2 + 3 × 4 = 14, NOT 20.
- ✗Thinking M always comes before D, or A always before S. Multiplication/Division are EQUAL priority (left to right). Same for Addition/Subtraction.
- ✗Forgetting that exponents apply before multiplication. 2 × 3² = 2 × 9 = 18, NOT 6² = 36.
- ✗Not working inside parentheses first. In 5(2 + 3), the parentheses are done FIRST: 5 × 5 = 25.
- ✗Applying exponents to everything inside parentheses incorrectly. (2+3)² = 5² = 25, NOT 2² + 3² = 13.
🌍 Real Life Example
Calculating Your Paycheck
Your weekly pay calculation: (40 × $20) + (5 × $20 × 1.5) - $150. Following PEMDAS: Parentheses first → (800) + (150) - 150. Then left to right: 800 + 150 - 150 = $800. Without PEMDAS, you'd get a wrong paycheck! The formula represents: regular hours × rate + overtime hours × rate × 1.5 overtime multiplier - deductions.
💡 Key Takeaway
PEMDAS gives math a universal rulebook: Parentheses first, then Exponents, then Multiplication/Division (left to right), then Addition/Subtraction (left to right). Remember: M and D are equal priority, as are A and S — when they're at the same level, just go left to right.