Fractions & Decimals

🎯 Learning Objective

Compare, add and convert fractions and decimals at exam speed.

💡 Concept

• Fraction = part/whole; same denominator → compare numerators
• Quick compare: cross-multiply a/b vs c/d → compare a·d vs b·c
• Decimal → fraction: 0.25 = 25/100 = 1/4
• Recurring: 0.333… = 3/9 = 1/3; 0.4̄5̄ = 45/99 = 5/11
• Learn the family: 1/2=0.5, 1/3≈0.333, 1/4=0.25, 1/5=0.2, 1/6≈0.1667, 1/8=0.125

🧮 Key Formulas

0.abab… = ab/99

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a/b vs c/d → compare ad vs bc

✏️ Easy Example

Q. Which is larger: 3/7 or 4/9?

1. Cross multiply: 3×9 = 27, 4×7 = 28
2. 28 > 27 → 4/9 is larger

Answer: 4/9

🇮🇳 Real-Life Example

Half a roti (1/2) vs three-eighths (3/8) of a roti — cross-multiplying settles who got more food.

📝 Exam-Level Example

Q. Convert 0.2̄7̄ (0.272727…) into a fraction.

1. Two repeating digits → denominator 99
2. = 27/99
3. = 3/11

Answer: 3/11

🪄 Memory Trick

Repeating block of n digits → put the block over n nines. Mixed (0.1[6]) → (16−1)/90 = 15/90 = 1/6.

⚠️ Common Mistakes

• ❌ Comparing fractions by looking at numerators only
• ❌ Writing 0.1666… as 16/99 (it's 1/6 — mixed recurring)

🏆 Exam Tips

• ✅ Memorise 1/n decimals till n=12
• ✅ Convert everything to decimals when options are far apart

📌 Summary

• Cross-multiply to compare
• Pure recurring → 9s; mixed → 9s and 0s
• Memorise the 1/n table