Mathematical Operations
About This Cheat Sheet
Your complete guide to Mathematical Operations for competitive exams. This cheat sheet helps you solve arithmetic problems, decode symbolic operations, and apply BODMAS rules quickly.
📐 Basic Operations & BODMAS
- Addition (+), Subtraction (-)
- Multiplication (×), Division (÷)
- BODMAS / PEMDAS Rules
- Brackets, Orders, Division, Multiplication
- Addition, Subtraction Priority
- Left-to-Right Rule
🔣 Symbolic & Coded Operations
- Symbols: +, -, ×, ÷, =, ≠, <, >
- ≤, ≥ (Less Than or Equal)
- Symbols Representing Operations
- Finding Meaning of Symbols
- Interchanging Symbols
- Decoding Coded Operations
🔄 Operation Substitution
- Replacing Operations
- Swapping Two Operations
- Operation with Digits
- Forming Numbers
- Operation with Fractions & Decimals
- Operation with Percentages
🧮 Advanced Operations
- Square Roots & Cube Roots
- Exponents & Powers
- Operation with Missing Numbers
- Operation with Conditions (AND, OR, NOT)
- Equation Solving
- Inequality Problems
🎯 Problem Types
- Find Missing Operator
- Find Correct Equation
- Find Correct Expression
- Find Value of Unknown
- Find Correct Relationship
- Operation-Based Puzzles
📚 Exam Categories
- SSC & Banking
- Railways & UPSC
- CAT & MBA Entrance
- Defence & Insurance
- Placement Tests
- Other Competitive Exams
🔴 B
Brackets
Highest Priority🟠 O
Orders
Powers & Roots🟡 D/M
Division / Multiplication
Left to Right🟢 A/S
Addition / Subtraction
Left to Right🔄 Common Symbol Substitutions
➕ Addition Substitutions
- + means - : 5 + 3 = 5 - 3 = 2
- + means × : 5 + 3 = 5 × 3 = 15
- + means ÷ : 5 + 3 = 5 ÷ 3
- + means ^ : 5 + 3 = 5³
➖ Subtraction Substitutions
- − means + : 5 − 3 = 5 + 3 = 8
- − means × : 5 − 3 = 5 × 3 = 15
- − means ÷ : 5 − 3 = 5 ÷ 3
- − means ^ : 5 − 3 = 5³
✖️ Multiplication Substitutions
- × means + : 5 × 3 = 5 + 3 = 8
- × means − : 5 × 3 = 5 − 3 = 2
- × means ÷ : 5 × 3 = 5 ÷ 3
- × means ^ : 5 × 3 = 5³
📊 Inequality Symbols
🔢 Operation Types
🧮 Quick Problem Solving Examples
📐 BODMAS Example
- Expression: 5 + 3 × 2
- Wrong: (5+3)×2 = 16
- Correct: 5 + (3×2) = 11
- Rule: Multiplication first
🔄 Symbol Substitution
- Given: + means −, − means +
- Expression: 5 + 3 − 2
- Replace: 5 − 3 + 2
- Result: 4
📊 Inequality Example
- Statement: A > B < C
- Conclusion: A > B, C > B
- Cannot conclude: A > C
- Rule: Only direct relationships
📋 Quick Reference: Operation Rules
📐 BODMAS Rules
- Brackets: ( ), { }, [ ]
- Orders: Powers (², ³), Roots (√)
- Division: ÷
- Multiplication: ×
- Addition: +
- Subtraction: −
🔄 Symbol Substitution
- + means − : + → −
- − means + : − → +
- × means ÷ : × → ÷
- ÷ means × : ÷ → ×
- Find the rule: Identify pattern
- Replace before: Solving
📊 Inequality Rules
- < : Less Than
- > : Greater Than
- ≤ : Less Than or Equal
- ≥ : Greater Than or Equal
- = : Equal To
- ≠ : Not Equal To
🎯 How to Solve Mathematical Operations Quickly
The Golden Rule: Always follow the BODMAS rule – Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. Without this, you'll get the wrong answer even if your calculations are correct.
Step-by-Step Approach:
- Step 1: Identify the operations in the expression
- Step 2: Check for any symbol substitutions or interchanges
- Step 3: Apply BODMAS rule strictly
- Step 4: Replace coded symbols with actual operations
- Step 5: Solve step by step to find the answer
Common Mistakes to Avoid:
- ❌ Ignoring BODMAS and solving left to right
- ❌ Confusing the order of operations
- ❌ Forgetting to replace coded symbols correctly
- ❌ Misinterpreting inequality symbols (<, >, ≤, ≥)
Proven Strategy: The "Replace & Solve" technique – first, replace all coded symbols with their actual operations. Then, write the expression clearly and apply BODMAS. This two-step approach eliminates confusion and ensures accuracy.
Memory Trick: Remember the BODMAS hierarchy:
- 🔴 B: Brackets (Highest Priority)
- 🟠 O: Orders (powers, square roots)
- 🟡 D/M: Division & Multiplication (Left to Right)
- 🟢 A/S: Addition & Subtraction (Left to Right)
💡 Pro Tips for Mathematical Operations
✅ Memorize BODMAS
BODMAS is the most important rule – memorize it perfectly
✅ Replace Before Solving
Always replace coded symbols first, then solve step by step
✅ Watch for Interchanges
When two operations are interchanged, swap them in the entire expression
✅ Verify with Options
If options are given, substitute the answer to verify
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