Percentage (சதவீதம்)

Percentage is the backbone of quantitative aptitude - it's like the universal language of mathematics that connects every topic! From profit & loss to simple interest, from data interpretation to ratio & proportion, percentage appears everywhere in competitive exams. Mastering percentage gives you the confidence and speed to tackle 60-70% of quantitative questions in TNPSC, SSC, Banking, and UPSC exams. It's not just about solving problems; it's about developing numerical intuition that makes complex calculations feel effortless.

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🧠 Foundation: Understanding the Core Concept

Percentage means "per hundred" or "out of 100." Think of it as a universal comparison tool 🧭 that helps us express any quantity as a fraction of 100, making it easy to compare different values regardless of their original size.

Real-world examples:

• 📱 Your phone battery: 80% means 80 out of 100 units of charge remaining
• 🎯 Exam scores: 85% means you got 85 marks out of every 100 possible marks
• 🛍️ Sale discounts: 25% off means you pay 75 out of every 100 rupees of original price
• 📈 Growth rates: Population increased by 15% means 15 more people for every 100 people

The Core Formula

English: Percentage = (Part/Whole) × 100 | Part = (Percentage/100) × Whole

Tamil: சதவீதம் = (பகுதி/முழுமை) × 100 | பகுதி = (சதவீதம்/100) × முழுமை

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Level 0: Formula Playground

Percentage Type	English Formula	Tamil Term	Example
Basic Percentage	(Part/Whole) × 100	அடிப்படை சதவீதம்	(25/100) × 100 = 25%
Finding Part	(% × Whole)/100	பகுதி கண்டறிதல்	(30% × 200) = 60
Finding Whole	(Part × 100)/%	முழுமை கண்டறிதல்	(75 × 100)/25% = 300
Percentage Change	((New-Old)/Old) × 100	மாற்றம் சதவீதம்	((120-100)/100) × 100 = 20%

Practice Drill 1: What is 15% of 240? Solution: (15/100) × 240 = 0.15 × 240 = 36

Practice Drill 2: 45 is what percent of 180? Solution: (45/180) × 100 = (1/4) × 100 = 25%

Practice Drill 3: If 30% of a number is 75, find the number. Solution: Number = (75 × 100)/30 = 7500/30 = 250

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⚙️ Unit Conversion Mastery (அலகு மாற்றம்)

Common conversion mistakes: Forgetting to multiply/divide by 100 when converting between percentage and decimal!

Key conversion factors:

• Percentage to Decimal: Divide by 100 (25% = 0.25)
• Decimal to Percentage: Multiply by 100 (0.75 = 75%)
• Fraction to Percentage: (Fraction × 100)% (3/4 = 75%)
• Common Percentages: 50% = 1/2, 25% = 1/4, 33⅓% = 1/3, 20% = 1/5

Conversion Practice Problem 1: Convert 0.125 to percentage Solution: 0.125 × 100 = 12.5%

Conversion Practice Problem 2: Express 7/8 as a percentage Solution: (7/8) × 100 = 0.875 × 100 = 87.5%

Conversion Practice Problem 3: If 65% of students passed, what fraction failed? Solution: Failed = 100% - 65% = 35% = 35/100 = 7/20

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Level 1: Problem Reading Strategy (கேள்வியை அணுகும் முறை)

G-F-V Framework:

• Given (கொடுக்கப்பட்டது): Identify the whole, part, or percentage mentioned
• Find (கண்டுபிடி): What percentage, part, or whole needs to be calculated
• Verify (சரிபார்): Check if answer makes logical sense (percentage should be reasonable)

Example 1: In a class of 40 students, 28 students passed the exam. What percentage of students passed?

• Given: Total students = 40, Passed students = 28
• Find: Percentage of students who passed
• Solution: Percentage = (28/40) × 100 = 70%
• Verify: 70% means more than half passed, which seems reasonable ✓

Example 2: A shirt costs ₹800. If there's a 15% discount, what is the sale price?

• Given: Original price = ₹800, Discount = 15%
• Find: Sale price after discount
• Solution: Discount amount = 15% of 800 = (15/100) × 800 = ₹120, Sale price = 800 - 120 = ₹680
• Verify: Sale price is less than original price ✓

Example 3: If 20% of a number is 60, find 35% of the same number.

• Given: 20% of number = 60
• Find: 35% of the same number
• Solution: Number = (60 × 100)/20 = 300, 35% of 300 = (35/100) × 300 = 105
• Verify: 35% should be larger than 20%, and 105 > 60 ✓

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Level 2: Pattern Recognition (கணக்குகளின் வகையை அறிதல்)

Pattern	English Keywords	Core Concept	Tamil Term
Basic Calculation	"what percent", "find %", "% of"	Direct percentage formula	அடிப்படை கணக்கீடு
Increase/Decrease	"increased by", "decreased by", "more/less"	Percentage change calculation	அதிகரிப்பு/குறைவு
Successive Changes	"first increase then decrease", "multiple changes"	Compound percentage changes	தொடர் மாற்றங்கள்
Comparison Problems	"A is x% more than B", "ratio comparison"	Relative percentage analysis	ஒப்பீட்டு பிரச்சினைகள்

Pattern 1: Basic Calculation (அடிப்படை கணக்கீடு)

Keywords: "What percent", "find the percentage", "% of a number" Core Insight: Use the fundamental percentage formula directly

Visual Diagram: Whole (100%) ────────────────────────

│ ├── a t (?% of Whole)

Formula Flow: Part/Whole × 10

Worked Example: In an office of 250 employees, 60 are women. What percentage are women? Solution:

• Given: Total = 250, Women = 60
• Apply formula: (60/250) × 100
• Simplify: (6/25) × 100 = 24%
• Check: 24% means about 1 in 4 employees are women ✓

Pattern 2: Increase/Decrease (அதிகரிப்பு/குறைவு)

Keywords: "increased by %", "decreased by %", "% more", "% less" Core Insight: Calculate change amount, then apply to original value

Visual Diagram: Original Value ──→ Change (±%) ──→ New Value

Increase: New = Original × (100 + %)100

Worked Example: A salary of ₹25,000 is increased by 12%. What is the new salary? Solution:

• Original salary = ₹25,000, Increase = 12%
• Method 1: Increase amount = 12% of 25,000 = 3,000, New salary = 25,000 + 3,000 = ₹28,000
• Method 2 (Shortcut): New salary = 25,000 × (100 + 12)/100 = 25,000 × 1.12 = ₹28,000

Pattern 3: Successive Changes (தொடர் மாற்றங்கள்)

Keywords: "first increased then decreased", "multiple percentage changes" Core Insight: Apply changes sequentially, not additively

Visual Diagram: Original → Change 1 → Intermediate → Change 2 → Final 100 → ±a% → X → ±b% → Result

Formula: Final = Original × (100±a)/100 × (100±b)/100

Worked Example: A number is first increased by 20%, then decreased by 15%. If the final result is 408, find the original number. Solution:

• Let original number = x
• After 20% increase: x × 1.20 = 1.2x
• After 15% decrease: 1.2x × 0.85 = 1.02x
• Given: 1.02x = 408
• Therefore: x = 408/1.02 = 400

Pattern 4: Comparison Problems (ஒப்பீட்டு பிரச்சினைகள்)

Keywords: "A is x% more than B", "A is x% of B", "ratio between" Core Insight: Identify the base value for comparison

Visual Diagram: Base Value B ────────────────── (100%) │ ├── Additional amount (x%) │ Compared Value A ──────────────── (100 + x%)

If A is x% more than B: A = B × (100 + x)/100

Worked Example: John's salary is 25% more than Peter's salary. If John earns ₹45,000, what does Peter earn? Solution:

• Let Peter's salary = P
• John's salary = P + 25% of P = P × (125/100) = 1.25P
• Given: 1.25P = 45,000
• Therefore: P = 45,000/1.25 = ₹36,000

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Level 3: Advanced Scenarios (சிக்கலான கணக்குகள்)

Advanced Pattern: Population/Election Problems These involve percentages of percentages and require careful identification of the base value.

Common Trap

In election problems, be careful about voter turnout vs. vote share. Total votes cast ≠ Total eligible voters!

Example 1: In an election, 60% of eligible voters voted. The winning candidate got 55% of votes cast and won by 3,600 votes. Find the number of eligible voters. Solution:

• Let eligible voters = E
• Votes cast = 60% of E = 0.6E
• Winner got 55% of votes cast = 0.55 × 0.6E = 0.33E
• Loser got 45% of votes cast = 0.45 × 0.6E = 0.27E
• Winning margin = 0.33E - 0.27E = 0.06E = 3,600
• Therefore: E = 3,600/0.06 = 60,000 eligible voters

Advanced Pattern: Mixture and Alligation Example 2: A solution contains 40% alcohol. How much water should be added to 50 liters of this solution to make it 25% alcohol? Solution:

• Current alcohol = 40% of 50 = 20 liters (this remains constant)
• Let x liters of water be added
• New solution volume = 50 + x liters
• New concentration: 20/(50 + x) = 25/100
• Cross multiply: 20 × 100 = 25(50 + x)
• 2000 = 1250 + 25x → x = 30 liters

Advanced Strategy

For mixture problems, remember that the actual amount of the substance remains constant - only the concentration changes when you add or remove the solvent.

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⚠️ Common Beginner Mistakes (பொதுவான தவறுகள்)

Mistake 1: Adding/Subtracting Successive Percentages ❌ Wrong: If price increases by 20% then decreases by 10%, net change = +20% - 10% = +10% ✅ Correct: New price = Original × 1.20 × 0.90 = Original × 1.08, so net increase = 8%

Mistake 2: Wrong Base for Percentage ❌ Wrong: "A is 50% more than B" means A = B + 50 ✅ Correct: A = B + 50% of B = B × 1.5

Mistake 3: Percentage vs Percentage Points Confusion ❌ Wrong: If pass rate increases from 60% to 80%, saying it increased by 20% ✅ Correct: It increased by 20 percentage points, or by (20/60) × 100 = 33.33%

Mistake 4: Decimal/Percentage Conversion Errors ❌ Wrong: Converting 0.05 as 5% (forgetting to multiply by 100) ✅ Correct: 0.05 = 0.05 × 100 = 5%

Mistake 5: Finding Wrong Percentage in Comparison ❌ Wrong: If A = 120 and B = 100, saying "A is 120% of B" ✅ Correct: A is 120% of B, which means A is 20% more than B

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🚀 Key Takeaways (முக்கிய குறிப்புகள்)

1. Master the Base Formula: Percentage = (Part/Whole) × 100 - everything else builds on this
2. Identify the Base: In comparison problems, clearly identify what the percentage is calculated "of"
3. Successive Changes Multiply: Never add/subtract successive percentage changes - multiply the multipliers
4. Convert Consistently: Work in either percentages or decimals throughout - don't mix mid-calculation
5. Verify with Logic: Your answer should pass the common sense test

Success Formula

Percentage Mastery = (Practice × Pattern Recognition) + (Speed × Accuracy) × Consistent Base Identification Remember: Percentages are just fractions in disguise - unmask them and they become your allies! 💪

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📝 Problems to Practice (பயிற்சிக் கேள்விகள்)

Pattern 1: Basic Calculation (அடிப்படை கணக்கீடு) - 3 Problems

Problem 1: If 35% of 1200 students are girls, how many boys are there in the school? Pattern Identification: Basic percentage calculation to find remaining part Solution:

• Given: Total students = 1200, Girls = 35%
• Step 1: Find number of girls → 35% of 1200 = (35/100) × 1200 = 420 girls
• Step 2: Find boys → Boys = Total - Girls = 1200 - 420 = 780 boys
• Step 3: Verification → 780/1200 × 100 = 65%, which matches 100% - 35% ✓ Answer: 780 boys

Problem 2: What percent of 2.5 kg is 750 grams? Pattern Identification: Finding percentage when part and whole are given in different units Solution:

• Given: Whole = 2.5 kg, Part = 750 grams
• Step 1: Convert to same units → 2.5 kg = 2500 grams
• Step 2: Apply percentage formula → (750/2500) × 100
• Step 3: Simplify → (3/10) × 100 = 30%
• Step 4: Verification → 30% of 2.5 kg = 0.75 kg = 750 grams ✓ Answer: 30%

Problem 3: If 16% of a number is 72, find 25% of the same number. Pattern Identification: Finding whole from given percentage, then calculating another percentage Solution:

• Given: 16% of number = 72
• Step 1: Find the number → Number = (72 × 100)/16 = 450
• Step 2: Find 25% of this number → 25% of 450 = (25/100) × 450 = 112.5
• Step 3: Verification → 16% of 450 = 72 ✓, 25% > 16% so 112.5 > 72 ✓ Answer: 112.5

Pattern 2: Increase/Decrease (அதிகரிப்பு/குறைவு) - 3 Problems

Problem 4: The price of petrol increased from ₹80 per liter to ₹92 per liter. Find the percentage increase. Pattern Identification: Percentage increase calculation from old to new value Solution:

• Given: Old price = ₹80, New price = ₹92
• Step 1: Find increase → Increase = 92 - 80 = ₹12
• Step 2: Apply percentage increase formula → (12/80) × 100
• Step 3: Calculate → (12/80) × 100 = 15%
• Step 4: Verification → 15% of 80 = 12, so new price = 80 + 12 = 92 ✓ Answer: 15%

Problem 5: After a 20% discount, a book costs ₹240. What was the original price? Pattern Identification: Finding original value from discounted value Solution:

• Given: Discounted price = ₹240, Discount = 20%
• Step 1: After 20% discount, customer pays 80% of original price
• Step 2: Set up equation → 80% of original price = 240
• Step 3: Find original price → Original = (240 × 100)/80 = ₹300
• Step 4: Verification → 20% of 300 = 60, so discounted price = 300 - 60 = 240 ✓ Answer: ₹300

Problem 6: A population of 50,000 increased by 8% in the first year and decreased by 5% in the second year. Find the population after two years. Pattern Identification: Successive percentage changes (increase then decrease) Solution:

• Given: Initial = 50,000, First year +8%, Second year -5%
• Step 1: After first year → 50,000 × (100 + 8)/100 = 50,000 × 1.08 = 54,000
• Step 2: After second year → 54,000 × (100 - 5)/100 = 54,000 × 0.95 = 51,300
• Step 3: Alternative method → 50,000 × 1.08 × 0.95 = 50,000 × 1.026 = 51,300
• Step 4: Net change = (51,300 - 50,000)/50,000 × 100 = 2.6% increase Answer: 51,300

Pattern 3: Successive Changes (தொடர் மாற்றங்கள்) - 3 Problems

Problem 7: The length of a rectangle is increased by 25% and its width is decreased by 20%. Find the percentage change in its area. Pattern Identification: Successive percentage changes affecting a derived quantity (area) Solution:

• Given: Length increases by 25%, Width decreases by 20%
• Step 1: Let original length = L, width = W, area = L × W
• Step 2: New length = L × 1.25, New width = W × 0.80
• Step 3: New area = (L × 1.25) × (W × 0.80) = L × W × 1.25 × 0.80 = L × W × 1.00
• Step 4: Percentage change = ((New - Original)/Original) × 100 = 0% Answer: 0% (no change)

Problem 8: A number is first increased by 40%, then the result is decreased by 30%. If the final number is 168, find the original number. Pattern Identification: Working backwards from final result through successive changes Solution:

• Given: Final result = 168, Changes: +40% then -30%
• Step 1: Let original number = x
• Step 2: After 40% increase → x × 1.40 = 1.4x
• Step 3: After 30% decrease → 1.4x × 0.70 = 0.98x
• Step 4: Set up equation → 0.98x = 168
• Step 5: Solve → x = 168/0.98 = approximately 171.43 Answer: ≈171.43

Problem 9: In three consecutive years, the population of a city changes by +10%, -5%, and +8% respectively. If the final population is 113,652, what was the initial population? Pattern Identification: Multiple successive changes with reverse calculation Solution:

• Given: Final population = 113,652, Changes: +10%, -5%, +8%
• Step 1: Let initial population = P
• Step 2: After three changes → P × 1.10 × 0.95 × 1.08 = P × 1.1286
• Step 3: Set up equation → P × 1.1286 = 113,652
• Step 4: Solve → P = 113,652/1.1286 = 100,000
• Step 5: Verification → 100,000 × 1.10 × 0.95 × 1.08 = 113,652 ✓ Answer: 100,000

Pattern 4: Comparison Problems (ஒப்பீட்டு பிரச்சினைகள்) - 3 Problems

Problem 10: Ram's salary is 20% more than Shyam's salary. If Ram saves 25% of his salary and Shyam saves 20% of his salary, and Ram saves ₹300 more than Shyam, find both their salaries. Pattern Identification: Comparison with additional conditions and equation setup Solution:

• Given: Ram = 120% of Shyam, Ram saves 25%, Shyam saves 20%, Ram saves ₹300 more
• Step 1: Let Shyam's salary = S, Ram's salary = 1.2S
• Step 2: Shyam's savings = 20% of S = 0.2S
• Step 3: Ram's savings = 25% of 1.2S = 0.3S
• Step 4: Set up equation → 0.3S - 0.2S = 300 → 0.1S = 300 → S = ₹3,000
• Step 5: Ram's salary = 1.2 × 3,000 = ₹3,600 Answer: Shyam: ₹3,000, Ram: ₹3,600

Problem 11: In an examination, 60% of students passed in Math, 70% passed in Science, and 40% passed in both subjects. If 200 students failed in both subjects, find the total number of students. Pattern Identification: Venn diagram problem with percentages and overlapping sets Solution:

• Given: Math pass = 60%, Science pass = 70%, Both pass = 40%, Both fail = 200
• Step 1: Students passing in at least one = Math + Science - Both = 60% + 70% - 40% = 90%
• Step 2: Students failing in both = 100% - 90% = 10%
• Step 3: Given that 10% = 200 students
• Step 4: Total students = (200 × 100)/10 = 2,000
• Step 5: Verification → 90% of 2000 = 1800 passed, 10% of 2000 = 200 failed ✓ Answer: 2,000 students

Problem 12: A's income is 25% less than B's income. B's income is 40% more than C's income. If C's income is ₹15,000, find A's income and express it as a percentage of C's income. Pattern Identification: Chain comparison through multiple relationships Solution:

• Given: A = B - 25% of B, B = C + 40% of C, C = ₹15,000
• Step 1: Find B's income → B = 140% of C = 1.4 × 15,000 = ₹21,000
• Step 2: Find A's income → A = 75% of B = 0.75 × 21,000 = ₹15,750
• Step 3: A as percentage of C → (15,750/15,000) × 100 = 105%
• Step 4: Verification → A is 105% of C, meaning A earns 5% more than C ✓ Answer: A's income = ₹15,750, which is 105% of C's income