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HCF and LCM of Fractions (பின்னங்களின் மீ.பொ.வ மற்றும் சி.பொ.ம)

Welcome beginners! 🎯 Understanding HCF and LCM of fractions is crucial for competitive exams as these concepts appear frequently in quantitative aptitude sections. Mastering these formulas helps you solve complex fraction problems, time and work questions, ratio and proportion problems, and many real-world scenarios involving fractional quantities. This topic builds a strong foundation for advanced mathematical reasoning! 💪

🧠 Foundation: Understanding the Core Concept (அடிப்படை கருத்துகள்)​

HCF (Highest Common Factor) of fractions is the largest fraction that divides all given fractions exactly. In Tamil: மீ.பொ.வ (மீப்பெரு பொது வகுப்பான்) - பின்னங்களை முழுமையாக வகுக்கும் மிகப்பெரிய பின்னம்.

LCM (Least Common Multiple) of fractions is the smallest fraction that is divisible by all given fractions. In Tamil: சி.பொ.ம (சிறுபொது மடங்கு) - அனைத்து பின்னங்களாலும் வகுபடும் மிகச்சிறிய பின்னம்.

Real-world examples:

  • 🍕 Sharing pizza slices: Finding equal portions (HCF)
  • ⏰ Time intervals: When events repeat together (LCM)
  • 📏 Measuring tape: Common measurement units (HCF/LCM)
The Core Formulas

HCF of fractions = HCF of numerators / LCM of denominators LCM of fractions = LCM of numerators / HCF of denominators

பின்னங்களின் மீ.பொ.வ = தொகையின் மீ.பொ.வ / பகுதியின் சி.பொ.ம பின்னங்களின் சி.பொ.ம = தொகையின் சி.பொ.ம / பகுதியின் மீ.பொ.வ

Level 0: Formula Playground​

OperationEnglish FormulaTamil Formula
HCF of a/b, c/dHCF(a,c) / LCM(b,d)மீ.பொ.வ(a,c) / சி.பொ.ம(b,d)
LCM of a/b, c/dLCM(a,c) / HCF(b,d)சி.பொ.ம(a,c) / மீ.பொ.வ(b,d)
Three fractionsSame pattern appliesஅதே முறை பொருந்தும்

Practice Drills:

Drill 1: Find HCF of 2/3 and 4/9

  • HCF of numerators: HCF(2,4) = 2
  • LCM of denominators: LCM(3,9) = 9
  • HCF = 2/9 ✅

Drill 2: Find LCM of 1/2 and 1/3

  • LCM of numerators: LCM(1,1) = 1
  • HCF of denominators: HCF(2,3) = 1
  • LCM = 1/1 = 1 ✅

Drill 3: Find HCF of 6/8 and 9/12

  • Simplify first: 3/4 and 3/4
  • HCF of numerators: HCF(3,3) = 3
  • LCM of denominators: LCM(4,4) = 4
  • HCF = 3/4 ✅

⚙️ Unit Conversion Mastery (அலகு மாற்றம்)​

Common conversion mistakes:

  • Mixing improper fractions with proper fractions
  • Not simplifying fractions before calculation
  • Confusing HCF and LCM formulas

Key conversion factors:

  • Mixed numbers → Improper fractions: 2¾ = 11/4
  • Decimals → Fractions: 0.25 = 1/4, 0.75 = 3/4
  • Percentages → Fractions: 25% = 1/4, 66.67% = 2/3

Conversion Practice Problems:

Problem 1: Convert and find HCF of 1.5 and 2.25

  • Convert: 1.5 = 3/2, 2.25 = 9/4
  • HCF = HCF(3,9)/LCM(2,4) = 3/4 = 0.75 ✅

Problem 2: Find LCM of 33.33% and 50%

  • Convert: 33.33% = 1/3, 50% = 1/2
  • LCM = LCM(1,1)/HCF(3,2) = 1/1 = 1 ✅

Problem 3: Find HCF of 2⅓ and 1¾

  • Convert: 2⅓ = 7/3, 1¾ = 7/4
  • HCF = HCF(7,7)/LCM(3,4) = 7/12 ✅

Level 1: Problem Reading Strategy (கேள்வியை அணுகும் முறை)​

G-F-V Framework (Given-Find-Verify):

  • Given: Identify the fractions and operation needed
  • Find: Apply correct formula and calculate step-by-step
  • Verify: Check if answer makes logical sense

Example 1 using G-F-V: Find HCF of 4/6 and 8/12.

Given (கொடுக்கப்பட்டது): Two fractions 4/6 and 8/12 Find (கண்டுபிடிக்க வேண்டியது): HCF of these fractions Solution:

  • Simplify: 4/6 = 2/3, 8/12 = 2/3
  • Since both are equal: HCF = 2/3 Verify (சரிபார்த்தல்): 2/3 divides both fractions exactly ✅

Example 2 using G-F-V: Find LCM of 2/5 and 3/10.

Given: Fractions 2/5 and 3/10 Find: LCM of these fractions
Solution:

  • LCM = LCM(2,3)/HCF(5,10) = 6/5 Verify: 6/5 ÷ (2/5) = 3, 6/5 ÷ (3/10) = 4 (both whole numbers) ✅

Example 3 using G-F-V: Find HCF of 9/16 and 12/20.

Given: Fractions 9/16 and 12/20
Find: HCF of these fractions Solution:

  • Simplify 12/20 = 3/5
  • HCF = HCF(9,3)/LCM(16,5) = 3/80 Verify: Both fractions divided by 3/80 give whole numbers ✅

Level 2: Pattern Recognition (கணக்குகளின் வகையை அறிதல்)​

PatternEnglish KeywordsCore ConceptTamil Term
Direct HCF/LCM"find HCF", "find LCM"Apply formulas directlyநேரடி கணக்கு
Mixed fractions"mixed numbers", "whole parts"Convert to improper firstகலப்பு பின்னம்
Three or more"multiple fractions", "several"Apply step by stepபல பின்னங்கள்
Word problems"sharing", "repeating", "common"Identify HCF/LCM contextவாழ்க்கை சூழல்

Pattern 1: Direct HCF/LCM (நேரடி கணக்கு) Keywords: "Find HCF of", "Calculate LCM of" Core insight: Apply formulas directly after simplifying fractions

Visual representation: Fraction 1: a/b Fraction 2: c/d ↓ ↓ Apply HCF/LCM formulas ↓ Final answer

Worked example: Find LCM of 5/8 and 7/12

  • Step 1: LCM of numerators = LCM(5,7) = 35
  • Step 2: HCF of denominators = HCF(8,12) = 4
  • Step 3: LCM = 35/4
  • Answer: 35/4 ✅

Pattern 2: Mixed Fractions (கலப்பு பின்னம்)
Keywords: "mixed numbers", "whole and fraction parts" Core insight: Convert to improper fractions first

Mixed fraction: 2¾ ↓ Convert: (2×4+3)/4 = 11/4 ↓ Apply HCF/LCM formulas

Worked example: Find HCF of 1½ and 2¼

  • Convert: 1½ = 3/2, 2¼ = 9/4
  • HCF = HCF(3,9)/LCM(2,4) = 3/4
  • Answer: 3/4 ✅

Pattern 3: Multiple Fractions (பல பின்னங்கள்) Keywords: "three fractions", "find common", "multiple values"
Core insight: Find HCF/LCM of first two, then with third

Fractions: a/b, c/d, e/f ↓ Find HCF/LCM of (a/b, c/d) = result₁ ↓ Find HCF/LCM of (result₁, e/f) = final answer

Worked example: Find HCF of 2/3, 4/9, and 8/27

  • Step 1: HCF(2/3, 4/9) = 2/9
  • Step 2: HCF(2/9, 8/27) = 2/27
  • Answer: 2/27 ✅

Level 3: Advanced Scenarios (சிக்கலான கணக்குகள்)​

Complex Scenario 1: Decimal and Fraction Mix When given decimals and fractions together, convert everything to fractions first.

Common Trap Alert!

Don't mix decimals and fractions directly. Always convert to common form first! தவறு: 0.5 மற்றும் 1/3 ஐ நேரடியாக கணக்கிடுதல் சரி: 0.5 = 1/2 ஆக மாற்றி பின் கணக்கிடுதல்

Example: Find LCM of 0.75, 2/3, and 5/6

  • Convert: 0.75 = 3/4
  • Find LCM of 3/4, 2/3, 5/6
  • LCM(3/4, 2/3) = 3/2, then LCM(3/2, 5/6) = 15/2
  • Answer: 15/2 = 7.5 ✅

Complex Scenario 2: Percentage Problems Convert percentages to fractions, then proceed.

Example: Find HCF of 25%, 40%, and 60%

  • Convert: 25% = 1/4, 40% = 2/5, 60% = 3/5
  • HCF(1/4, 2/5) = 1/20, then HCF(1/20, 3/5) = 1/20
  • Answer: 1/20 = 5% ✅
Advanced Strategy

For complex problems: Simplify → Convert → Calculate → Verify மேம்பட்ட உத்தி: எளிமைபடுத்து → மாற்று → கணக்கிடு → சரிபார் ✅

⚠️ Common Beginner Mistakes (பொதுவான தவறுகள்)​

Mistake 1: Formula Confusion ❌ Wrong: Using HCF formula for LCM problems ✅ Correct: HCF = HCF(numerators)/LCM(denominators), LCM = LCM(numerators)/HCF(denominators)

Mistake 2: Not Simplifying First
❌ Wrong: Finding HCF of 6/8 and 9/12 without simplifying ✅ Correct: Simplify to 3/4 and 3/4, then find HCF = 3/4

Mistake 3: Mixed Number Errors ❌ Wrong: Treating 2¼ as 2/4 ✅ Correct: Convert 2¼ = (2×4+1)/4 = 9/4

Mistake 4: Unit Inconsistency ❌ Wrong: Mixing percentages, decimals, and fractions without conversion ✅ Correct: Convert all to same form (preferably fractions) first

Mistake 5: Calculation Errors in Basic HCF/LCM ❌ Wrong: HCF(12,18) = 4
✅ Correct: HCF(12,18) = 6, LCM(12,18) = 36

🚀 Key Takeaways (முக்கிய குறிப்புகள்)​

  1. Master the Core Formulas: HCF = HCF(numerators)/LCM(denominators), LCM = LCM(numerators)/HCF(denominators) 📝

  2. Always Simplify First: Convert mixed numbers, reduce fractions to lowest terms before applying formulas 🔄

  3. Use G-F-V Strategy: Given-Find-Verify approach prevents calculation errors and ensures logical answers ✅

  4. Convert Consistently: Change decimals, percentages, and mixed numbers to proper fractions for easier calculation 🔢

  5. Practice Pattern Recognition: Identify problem types quickly to choose the right approach and avoid common traps 🎯

Memory Formula

"Simplify Smart, Calculate Clear, Verify Victory!"
"எளிதாக்கு, சரியாக கணக்கிடு, வெற்றி உறுதி செய்!" 🏆

📝 Problems to Practice (பயிற்சிக் கேள்விகள்)​

Pattern 1: Direct HCF/LCM Problems (நேரடி கணக்குகள்)​

Problem 1: Find the HCF of 3/5 and 9/20. Pattern Identification: Direct HCF calculation with proper fractions Solution:

  • HCF of numerators: HCF(3,9) = 3
  • LCM of denominators: LCM(5,20) = 20
  • HCF = 3/20 Answer: 3/20

Problem 2: Calculate the LCM of 2/7 and 4/21. Pattern Identification: Direct LCM calculation with related denominators
Solution:

  • LCM of numerators: LCM(2,4) = 4
  • HCF of denominators: HCF(7,21) = 7
  • LCM = 4/7
    Answer: 4/7

Problem 3: Find the HCF of 5/12 and 15/16. Pattern Identification: Direct HCF with different denominators Solution:

  • HCF of numerators: HCF(5,15) = 5
  • LCM of denominators: LCM(12,16) = 48
  • HCF = 5/48 Answer: 5/48

Pattern 2: Mixed Number Problems (கலப்பு பின்னங்கள்)​

Problem 4: Find the LCM of 1¾ and 2⅖.
Pattern Identification: Mixed numbers requiring conversion to improper fractions Solution:

  • Convert: 1¾ = 7/4, 2⅖ = 12/5
  • LCM of numerators: LCM(7,12) = 84
  • HCF of denominators: HCF(4,5) = 1
  • LCM = 84/1 = 84 Answer: 84

Problem 5: Calculate the HCF of 2⅔ and 1⅙. Pattern Identification: Mixed numbers with related fractional parts Solution:

  • Convert: 2⅔ = 8/3, 1⅙ = 7/6
  • HCF of numerators: HCF(8,7) = 1
  • LCM of denominators: LCM(3,6) = 6
  • HCF = 1/6 Answer: 1/6

Problem 6: Find the LCM of 3½ and 2¼. Pattern Identification: Mixed numbers with quarter and half relationships
Solution:

  • Convert: 3½ = 7/2, 2¼ = 9/4
  • LCM of numerators: LCM(7,9) = 63
  • HCF of denominators: HCF(2,4) = 2
  • LCM = 63/2 Answer: 63/2

Pattern 3: Multiple Fraction Problems (பல பின்னங்கள்)​

Problem 7: Find the HCF of 2/3, 4/9, and 8/27. Pattern Identification: Three fractions with geometric progression relationship
Solution:

  • Step 1: HCF(2/3, 4/9) = HCF(2,4)/LCM(3,9) = 2/9
  • Step 2: HCF(2/9, 8/27) = HCF(2,8)/LCM(9,27) = 2/27
  • Final HCF = 2/27 Answer: 2/27

Problem 8: Calculate the LCM of 1/2, 1/3, and 1/6. Pattern Identification: Unit fractions with common denominators Solution:

  • Step 1: LCM(1/2, 1/3) = LCM(1,1)/HCF(2,3) = 1/1 = 1
  • Step 2: LCM(1, 1/6) = LCM(1,1)/HCF(1,6) = 1/1 = 1
  • Final LCM = 1
    Answer: 1

Problem 9: Find the HCF of 6/8, 9/12, and 15/20.
Pattern Identification: Three fractions that can be simplified first Solution:

  • Simplify: 6/8 = 3/4, 9/12 = 3/4, 15/20 = 3/4
  • All fractions are equal, so HCF = 3/4 Answer: 3/4

Pattern 4: Word Problems and Applications (வாழ்க்கை சூழல் கணக்குகள்)​

Problem 10: Three friends can complete tasks in 2/3 hour, 3/4 hour, and 5/6 hour respectively. What is the HCF of their completion times? Pattern Identification: Real-world application requiring HCF for common time intervals Solution:

  • Find HCF of 2/3, 3/4, and 5/6
  • Step 1: HCF(2/3, 3/4) = HCF(2,3)/LCM(3,4) = 1/12
  • Step 2: HCF(1/12, 5/6) = HCF(1,5)/LCM(12,6) = 1/12
  • HCF = 1/12 hour = 5 minutes Answer: 1/12 hour

Problem 11: A recipe calls for 3/4 cup flour and 5/8 cup sugar. What is the LCM of these measurements for scaling up the recipe? Pattern Identification: Practical application requiring LCM for recipe scaling
Solution:

  • Find LCM of 3/4 and 5/8
  • LCM of numerators: LCM(3,5) = 15
  • HCF of denominators: HCF(4,8) = 4
  • LCM = 15/4 cups
    Answer: 15/4 cups

Problem 12: Two machines produce widgets in cycles of 2/5 hour and 3/10 hour. When will they complete cycles together? Find the LCM. Pattern Identification: Synchronization problem requiring LCM calculation Solution:

  • Find LCM of 2/5 and 3/10
  • LCM of numerators: LCM(2,3) = 6
  • HCF of denominators: HCF(5,10) = 5
  • LCM = 6/5 hours = 1⅕ hours = 1 hour 12 minutes Answer: 6/5 hours