Lesson 1.4: Order of Operations & PEMDAS

📘 GMAT Mastery Lesson 1.4: Order of Operations & PEMDAS

🎯 Learning Objective

Master the correct order of operations (PEMDAS) to eliminate calculation errors and solve complex multi-step expressions with confidence on the GMAT.

🧠 Core Concept

The GMAT frequently tests order of operations in both obvious and subtle ways. A single misstep can turn a correct approach into a wrong answer. The key is systematic application of PEMDAS with attention to common pitfalls.

The PEMDAS Framework

PEMDAS = Parentheses → Exponents → Multiplication/Division → Addition/Subtraction

Priority Operation Key Rule Example
1 Parentheses Innermost first (2 + 3) × 4 = 5 × 4 = 20
2 Exponents Include roots 23 = 8, √16 = 4
3 M/D Left to right 8 ÷ 4 × 2 = 2 × 2 = 4
4 A/S Left to right 10 - 3 + 2 = 7 + 2 = 9

🔑 Critical Rule: Left-to-Right Evaluation

When operations have equal priority (M/D or A/S), always work left to right.

Common Mistake: Assuming multiplication always comes before division

  • ❌ Wrong: 12 ÷ 3 × 2 = 12 ÷ 6 = 2
  • ✅ Correct: 12 ÷ 3 × 2 = 4 × 2 = 8

📊 Worked Example: Step-by-Step Breakdown

Problem: Evaluate 6 + 12 ÷ (3 × 2)2 - 5

Solution Process:

Step 1: Parentheses First
6 + 12 ÷ (3 × 2)2 - 5
= 6 + 12 ÷ (6)2 - 5

Step 2: Exponents
= 6 + 12 ÷ 36 - 5

Step 3: Division
= 6 + 1/3 - 5 (or 6 + 0.333... - 5)

Step 4: Addition/Subtraction (Left to Right)
If using fractions: = 18/3 + 1/3 - 15/3
= 19/3 - 15/3 = 4/3

✅ Final Answer: 4/3 or approximately 1.33

⚠️ GMAT-Specific Traps & Solutions

Trap Example Wrong Approach Correct Approach
Order Confusion 8 ÷ 4 × 2 8 ÷ (4 × 2) = 1 8 ÷ 4 × 2 = 2 × 2 = 4
Negative Exponents -32 (-3)2 = 9 -(32) = -9
Nested Parentheses 2[(3+1) × 2] Skip inner parentheses (3+1) = 4 first, then 4 × 2 = 8, then 2 × 8 = 16
Fraction Operations (8/2) × 3 or 8/2 × 3 8 / (2 × 3) = 8/6 = 4/3 (8/2) × 3 = 4 × 3 = 12

🎯 Pro Tip: Visual Organization

Use your scratch paper strategically:

  • Box exponents and parentheses
  • Circle fractions to keep numerators/denominators clear
  • Draw arrows to track left-to-right operations
  • Number your steps to avoid skipping

🏃‍♂️ Speed Strategies for GMAT Success

Mental Math Shortcuts:

  1. Look for patterns: 24 = 16, 33 = 27, 42 = 16
  2. Simplify fractions early: 12/4 = 3 before continuing
  3. Use estimation: If answer choices are far apart, approximate

Time-Saving Techniques:

  • Scan the entire expression before starting
  • Identify the most complex part (usually parentheses or exponents)
  • Work systematically – don’t jump around

📝 Targeted Practice Set

Level 1: Foundation

  1. 18 ÷ 3 + 22
  2. 4 + (5 - 3)2 × 2
  3. 6 × 2 - 3 + 4 ÷ 2

Level 2: GMAT-Style

  1. 24/(3 × 2) + 42 - 10 (Note: I added parentheses for clarity based on typical GMAT presentation for such denominators)
  2. -23 + (4 - 1)2 ÷ 3
  3. 5 × 3 ÷ 15 + 23 - 6

Level 3: Challenge

  1. ( (2 + 3)2 / 5 ) - 3 × 2 + √16 (Parentheses added for clarity)
  2. 2[3 + 4(2 - 1)] ÷ (32 - 7)

Solutions:

  1. 18 ÷ 3 + 22 = 6 + 4 = 10
  2. 4 + (5 - 3)2 × 2 = 4 + 22 × 2 = 4 + 4 × 2 = 4 + 8 = 12
  3. 6 × 2 - 3 + 4 ÷ 2 = 12 - 3 + 2 = 9 + 2 = 11
  4. 24/6 + 16 - 10 = 4 + 16 - 10 = 20 - 10 = 10
  5. -23 + (4 - 1)2 ÷ 3 = -8 + 32 ÷ 3 = -8 + 9 ÷ 3 = -8 + 3 = -5
  6. 5 × 3 ÷ 15 + 23 - 6 = 15 ÷ 15 + 8 - 6 = 1 + 8 - 6 = 9 - 6 = 3
  7. ( (5)2 / 5 ) - 6 + 4 = (25/5) - 6 + 4 = 5 - 6 + 4 = -1 + 4 = 3
  8. 2[3 + 4(1)] ÷ (9 - 7) = 2[3 + 4] ÷ 2 = 2[7] ÷ 2 = 14 ÷ 2 = 7

🎯 Key Takeaways for GMAT Success

Essential Rules to Remember:

  • PEMDAS is non-negotiable – follow it exactly
  • Left-to-right for equal operations (M/D and A/S)
  • Parentheses include all grouping symbols: (), [], {}
  • Exponents come before negative signs unless parentheses dictate otherwise (e.g., -32 vs (-3)2)

Strategic Approach:

  1. Scan the problem for complexity indicators
  2. Use scratch paper systematically
  3. Double-check negative signs and exponents
  4. Verify your final answer makes sense in context

Red Flags to Watch For:

  • Multiple operations without clear grouping
  • Negative numbers with exponents
  • Fractions mixed with other operations
  • Time pressure leading to shortcuts

Remember: The GMAT rewards accuracy over speed. A systematic approach to order of operations will save you points and build confidence for more complex quantitative problems.

Mastered PEMDAS? You’re building a rock-solid foundation for GMAT Quant! Next up: Crucial diagnostic assessment.