Simplification (சுருக்குதல், எளிமைப்படுத்துதல்)

Simplification is the foundation of all quantitative aptitude topics. It's like learning to walk before you run. Every competitive exam - whether TNPSC, SSC, Banking, or UPSC - tests your ability to quickly and accurately simplify complex mathematical expressions. Mastering simplification gives you the speed and accuracy needed to tackle advanced topics like percentages, profit & loss, and data interpretation with confidence.

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🧠 Foundation: Understanding the Core Concept (அடிப்படை கருத்துகள்)

Simplification means reducing a complex mathematical expression to its simplest form by following the correct order of operations. Think of it like following a recipe 👨‍🍳 - you must add ingredients in the right sequence to get the perfect dish!

Real-world examples:

• 🧮 Calculating your monthly budget: Income - (Rent + Food + Transport)
• 🛒 Shopping total: (Item1 × Quantity1) + (Item2 × Quantity2) - Discount
• 📊 Performance score: (Test1 + Test2 + Test3) ÷ 3

The Core Formula

BODMAS Rule: Brackets → Orders (Powers/Roots) → Division → Multiplication → Addition → Subtraction

Tamil: வட்ட அடைப்பு → அடுக்குகள் → வகுத்தல் → பெருக்கல் → கூட்டல் → கழித்தல்

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Level 0: Formula Playground

Operation	English Symbol	Tamil Term	Priority
Brackets	( ), [ ],	அடைப்பு குறிகள்	1 (Highest)
Orders/Powers	x², √x, x³	அடுக்குகள்	2
Division	÷, /	வகுத்தல்	3
Multiplication	×, *	பெருக்கல்	4
Addition	+	கூட்டல்	5
Subtraction	-	கழித்தல்	6 (Lowest)

Practice Drill 1: 8 + 2 × 3 Solution: First multiply: 2 × 3 = 6, then add: 8 + 6 = 14

Practice Drill 2: (10 - 6) × 3 + 5 Solution: Brackets first: (10 - 6) = 4, multiply: 4 × 3 = 12, add: 12 + 5 = 17

Practice Drill 3: 36 ÷ 6 + 2 × 4 Solution: Division and multiplication first: 36 ÷ 6 = 6, 2 × 4 = 8, then add: 6 + 8 = 14

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⚙️ Unit Conversion Mastery (அலகு மாற்றம்)

Common conversion mistakes: Mixing fractions with decimals without proper conversion!

Key conversion factors:

• Fraction to Decimal: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75
• Percentage: 1% = 0.01, 50% = 1/2 = 0.5
• Powers: 2² = 4, 3² = 9, 5² = 25, 10² = 100

Conversion Practice Problem 1: Convert 3/4 + 0.25 to decimal form Solution: 3/4 = 0.75, so 0.75 + 0.25 = 1.00

Conversion Practice Problem 2: Simplify 25% of 80 + 1/5 of 100 Solution: 25% = 0.25, so 0.25 × 80 = 20; 1/5 of 100 = 20; Answer: 20 + 20 = 40

Conversion Practice Problem 3: Calculate 2² + √16 - 50% Solution: 2² = 4, √16 = 4, 50% = 0.5; Answer: 4 + 4 - 0.5 = 7.5

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Level 1: Problem Reading Strategy (கேள்வியை அணுகும் முறை)

G-F-V Framework:

• Given (கொடுக்கப்பட்டது): What expression needs to be simplified?
• Find (கண்டுபிடி): The simplified numerical value
• Verify (சரிபார்): Check using reverse calculation or estimation

Example 1: Simplify: 15 + 3 × 2 - 8 ÷ 4

• Given: 15 + 3 × 2 - 8 ÷ 4
• Find: Apply BODMAS to get the final answer
• Solution: 3 × 2 = 6, 8 ÷ 4 = 2, so 15 + 6 - 2 = 19
• Verify: 19 seems reasonable (between 10-25)

Example 2: Calculate: (20 - 8) ÷ 3 + 5 × 2

• Given: (20 - 8) ÷ 3 + 5 × 2
• Find: Simplified value
• Solution: (20 - 8) = 12, 12 ÷ 3 = 4, 5 × 2 = 10, so 4 + 10 = 14
• Verify: 14 ✓

Example 3: Simplify: 2³ + √25 - 4 × 2

• Given: 2³ + √25 - 4 × 2
• Find: Final simplified value
• Solution: 2³ = 8, √25 = 5, 4 × 2 = 8, so 8 + 5 - 8 = 5
• Verify: 5 ✓

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Level 2: Pattern Recognition (கணக்குகளின் வகையை அறிதல்)

Pattern	English Keywords	Core Concept	Tamil Term
Basic BODMAS	+, -, ×, ÷ with brackets	Order of operations	அடிப்படை வரிசைமுறை
Powers & Roots	x², √x, cube root	Exponential operations	அடுக்கு வரிசைமுறை
Fraction Mix	1/2, 3/4, decimals	Mixed number operations	கலப்பு எண் கணக்குகள்
Nested Operations	Multiple brackets	Complex grouping	பல அடுக்கு அடைப்புகள்

Pattern 1: Basic BODMAS (அடிப்படை BODMAS)

Keywords: Simple arithmetic with +, -, ×, ÷ and basic brackets Core Insight: Always resolve brackets first, then follow BODMAS sequence

Visual Diagram: Expression: 10 + 2 × (8 - 3) - 4 ÷ 2

Step 1: Brackets → 10 + 2 × 5 - 4 ÷ 2 Step 2: Multiply → 10 + 10 - 4 ÷ 2 Step 3: Divide → 10 + 10 - 2 Step 4: Add/Subtract→ 20 - 2 = 18

Worked Example: 5 × 3 + 18 ÷ 6 - 2 Solution:

• Multiply first: 5 × 3 = 15
• Divide: 18 ÷ 6 = 3
• Now: 15 + 3 - 2 = 18 - 2 = 16

Pattern 2: Powers & Roots (அடுக்கு வரிசைமுறை)

Keywords: x², √x, x³, cube roots Core Insight: Powers and roots are resolved immediately after brackets

Visual Diagram: Expression: 3² + √16 × 2 - 8

Step 1: Powers/Roots → 9 + 4 × 2 - 8 Step 2: Multiply → 9 + 8 - 8 Step 3: Add/Subtract → 17 - 8 = 9

Worked Example: 2³ × 3 + √36 ÷ 2 Solution:

• Powers/roots first: 2³ = 8, √36 = 6
• Expression becomes: 8 × 3 + 6 ÷ 2
• Multiply/divide: 24 + 3 = 27

Pattern 3: Fraction Mix (கலப்பு எண் கணக்குகள்)

Keywords: 1/2, 3/4, 0.25, percentages mixed with whole numbers Core Insight: Convert all to same form (decimal or fraction) before calculation

Visual Diagram: Expression: 1/2 of 24 + 0.25 × 8 - 3

Step 1: Convert → 0.5 × 24 + 0.25 × 8 - 3 Step 2: Multiply → 12 + 2 - 3 Step 3: Add/Subtract → 14 - 3 = 11

Worked Example: 3/4 × 16 + 25% of 20 - 5 Solution:

• Convert: 3/4 = 0.75, 25% = 0.25
• Calculate: 0.75 × 16 = 12, 0.25 × 20 = 5
• Final: 12 + 5 - 5 = 12

Pattern 4: Nested Operations (பல அடுக்கு அடைப்புகள்)

Keywords: Multiple brackets like [( )], complex groupings Core Insight: Work from innermost bracket outward

Visual Diagram: Expression: 2 × [15 - (3 + 2) × 2] + 4

Step 1: Inner brackets → 2 × [15 - 5 × 2] + 4 Step 2: Multiplication → 2 × [15 - 10] + 4 Step 3: Outer brackets → 2 × 5 + 4 Step 4: Final → 10 + 4 = 14

Worked Example: 3 + 2 × [20 - (4 + 6) ÷ 2] Solution:

• Inner brackets: (4 + 6) = 10
• Division: 10 ÷ 2 = 5
• Outer brackets: [20 - 5] = 15
• Multiplication: 2 × 15 = 30
• Addition: 3 + 30 = 33

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Level 3: Advanced Scenarios (சிக்கலான கணக்குகள்)

Advanced Pattern: Mixed Operations with Decimals When dealing with complex expressions involving multiple decimal operations, always maintain precision and round only at the final step.

Common Trap

Don't round intermediate results! This leads to cumulative errors. Keep full precision until the final answer.

Example 1: 2.5 × 1.6 + 3.75 ÷ 1.25 - 0.8² Solution:

• Powers first: 0.8² = 0.64
• Multiply: 2.5 × 1.6 = 4.0
• Divide: 3.75 ÷ 1.25 = 3.0
• Final: 4.0 + 3.0 - 0.64 = 6.36

Advanced Pattern: Percentage with Fraction Operations Example 2: 40% of (3/4 × 80) + 150% of 20 Solution:

• Inner calculation: 3/4 × 80 = 60
• First term: 40% of 60 = 0.40 × 60 = 24
• Second term: 150% of 20 = 1.50 × 20 = 30
• Final: 24 + 30 = 54

Advanced Strategy

For percentage problems, convert percentages to decimals early: 40% → 0.4, 150% → 1.5

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⚠️ Common Beginner Mistakes (பொதுவான தவறுகள்)

Mistake 1: Wrong Order of Operations ❌ Wrong: 2 + 3 × 4 = (2 + 3) × 4 = 20 ✅ Correct: 2 + 3 × 4 = 2 + 12 = 14

Mistake 2: Bracket Confusion
❌ Wrong: 2 × (5 + 3 → solving without closing bracket ✅ Correct: 2 × (5 + 3) = 2 × 8 = 16

Mistake 3: Decimal/Fraction Mixing ❌ Wrong: 1/2 + 0.3 = 1/2 + 3/10 = 4/12 (incorrect conversion) ✅ Correct: 1/2 + 0.3 = 0.5 + 0.3 = 0.8

Mistake 4: Power Calculation Errors ❌ Wrong: 2³ = 2 × 3 = 6
✅ Correct: 2³ = 2 × 2 × 2 = 8

Mistake 5: Rounding Too Early ❌ Wrong: 2.67 × 3.14 ≈ 2.7 × 3.1 = 8.37 ✅ Correct: 2.67 × 3.14 = 8.3838, then round to 8.38

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🚀 Key Takeaways (முக்கிய குறிப்புகள்)

1. BODMAS is King: Always follow Brackets → Orders → Division/Multiplication → Addition/Subtraction
2. Convert Consistently: Keep fractions as fractions or decimals as decimals throughout the problem
3. Work Inside-Out: For nested brackets, start from the innermost and work outward
4. Precision Matters: Don't round intermediate steps; round only the final answer
5. Practice Daily: Speed comes from muscle memory - solve 10 problems daily

Motivation Formula

Success = (Practice × Accuracy) ÷ Time + Confidence² Keep practicing, and you'll master simplification! 💪

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📝 Problems to Practice (பயிற்சிக் கேள்விகள்)

Pattern 1: Basic BODMAS (அடிப்படை BODMAS) - 3 Problems

Problem 1: Simplify: 8 + 12 ÷ 4 - 2 × 3
Pattern Identification: Basic BODMAS with division, multiplication, addition, and subtraction Solution:

• Step 1: Division first → 12 ÷ 4 = 3
• Step 2: Multiplication → 2 × 3 = 6
• Step 3: Expression becomes → 8 + 3 - 6
• Step 4: Left to right → 11 - 6 = 5 Answer: 5

Problem 2: Calculate: (15 - 7) × 3 + 18 ÷ 6 Pattern Identification: Basic BODMAS with brackets, multiplication, division, and addition
Solution:

• Step 1: Brackets first → (15 - 7) = 8
• Step 2: Multiplication → 8 × 3 = 24
• Step 3: Division → 18 ÷ 6 = 3
• Step 4: Addition → 24 + 3 = 27 Answer: 27

Problem 3: Simplify: 36 - 4 × 3 + 20 ÷ 5 Pattern Identification: Basic BODMAS with multiplication, division, subtraction, and addition Solution:

• Step 1: Multiplication → 4 × 3 = 12
• Step 2: Division → 20 ÷ 5 = 4
• Step 3: Expression becomes → 36 - 12 + 4
• Step 4: Left to right → 24 + 4 = 28 Answer: 28

Pattern 2: Powers & Roots (அடுக்கு வரிசைமுறை) - 3 Problems

Problem 4: Calculate: 3² + √49 - 2³ ÷ 4 Pattern Identification: Powers and roots with basic operations Solution:

• Step 1: Powers and roots → 3² = 9, √49 = 7, 2³ = 8
• Step 2: Division → 8 ÷ 4 = 2
• Step 3: Expression becomes → 9 + 7 - 2
• Step 4: Left to right → 16 - 2 = 14 Answer: 14

Problem 5: Simplify: 5² - 3 × √16 + 4 Pattern Identification: Powers, roots, and multiplication mixed Solution:

• Step 1: Powers and roots → 5² = 25, √16 = 4
• Step 2: Multiplication → 3 × 4 = 12
• Step 3: Expression becomes → 25 - 12 + 4
• Step 4: Left to right → 13 + 4 = 17 Answer: 17

Problem 6: Calculate: √64 ÷ 2 + 3³ - 10 Pattern Identification: Roots, powers, division, and subtraction Solution:

• Step 1: Roots and powers → √64 = 8, 3³ = 27
• Step 2: Division → 8 ÷ 2 = 4
• Step 3: Expression becomes → 4 + 27 - 10
• Step 4: Left to right → 31 - 10 = 21 Answer: 21

Pattern 3: Fraction Mix (கலப்பு எண் கணக்குகள்) - 3 Problems

Problem 7: Simplify: 1/2 of 18 + 3/4 × 16 - 0.5 × 10 Pattern Identification: Mixed fractions and decimals Solution:

• Step 1: Convert to consistent form → 1/2 = 0.5, 3/4 = 0.75
• Step 2: Calculate each term → 0.5 × 18 = 9, 0.75 × 16 = 12, 0.5 × 10 = 5
• Step 3: Expression becomes → 9 + 12 - 5
• Step 4: Left to right → 21 - 5 = 16 Answer: 16

Problem 8: Calculate: 25% of 40 + 1/5 × 30 - 0.25 × 12
Pattern Identification: Percentages, fractions, and decimals mixed Solution:

• Step 1: Convert percentages → 25% = 0.25, 1/5 = 0.2
• Step 2: Calculate each term → 0.25 × 40 = 10, 0.2 × 30 = 6, 0.25 × 12 = 3
• Step 3: Expression becomes → 10 + 6 - 3
• Step 4: Left to right → 16 - 3 = 13 Answer: 13

Problem 9: Simplify: 2/3 × 15 + 50% of 24 - 3/8 of 16 Pattern Identification: Multiple fraction and percentage operations
Solution:

• Step 1: Convert to decimals → 2/3 = 0.667, 50% = 0.5, 3/8 = 0.375
• Step 2: Calculate each term → 0.667 × 15 = 10, 0.5 × 24 = 12, 0.375 × 16 = 6
• Step 3: Expression becomes → 10 + 12 - 6
• Step 4: Left to right → 22 - 6 = 16 Answer: 16

Pattern 4: Nested Operations (பல அடுக்கு அடைப்புகள்) - 3 Problems

Problem 10: Calculate: 5 + 3 × [24 - (8 + 4) ÷ 3] Pattern Identification: Nested brackets with multiple operations Solution:

• Step 1: Inner brackets → (8 + 4) = 12
• Step 2: Division → 12 ÷ 3 = 4
• Step 3: Outer brackets → [24 - 4] = 20
• Step 4: Multiplication → 3 × 20 = 60
• Step 5: Addition → 5 + 60 = 65 Answer: 65

Problem 11: Simplify: 2 × [(15 - 3) ÷ 4 + 6] - 8 Pattern Identification: Complex nested operations with division and addition Solution:

• Step 1: Inner brackets → (15 - 3) = 12
• Step 2: Division → 12 ÷ 4 = 3
• Step 3: Addition in brackets → 3 + 6 = 9
• Step 4: Multiplication → 2 × 9 = 18
• Step 5: Subtraction → 18 - 8 = 10 Answer: 10

Problem 12: Calculate: 48 ÷ [3 × (2² + 4) - 6] + 5 Pattern Identification: Deeply nested operations with powers Solution:

• Step 1: Power in inner brackets → 2² = 4
• Step 2: Inner brackets → (4 + 4) = 8
• Step 3: Multiplication → 3 × 8 = 24
• Step 4: Outer brackets → [24 - 6] = 18
• Step 5: Division → 48 ÷ 18 = 8/3 = 2.67
• Step 6: Addition → 2.67 + 5 = 7.67 Answer: 7.67