Half-Yearly & Quarterly Compounding

🎯 Learning Objective

Adjust rate and time when interest is compounded half-yearly or quarterly.

💡 Concept

• Half-yearly compounding → rate becomes r/2, periods become 2n
• A = P(1 + r/200)^(2n) for half-yearly
• Quarterly compounding → rate r/4, periods 4n
• 9 months quarterly = 3 periods; 18 months half-yearly = 3 periods
• More frequent compounding → slightly higher interest for the same nominal rate

🧮 Key Formulas

Half-yearly: A = P(1 + r/200)^(2n)

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Quarterly: A = P(1 + r/400)^(4n)

✏️ Easy Example

Q. Find the amount on ₹8,000 at 10% per annum compounded half-yearly for 1 year.

1. Rate = 5% per half-year, periods = 2
2. 8000 × 1.05 = 8400
3. 8400 × 1.05 = 8820

Answer: ₹8,820

🇮🇳 Real-Life Example

Bank FDs actually compound quarterly — an '8% p.a.' FD really adds 2% every 3 months, which is why the maturity slip shows a little extra.

📝 Exam-Level Example

Q. Find the compound interest on ₹10,000 at 20% per annum for 9 months, compounded quarterly.

1. Rate = 20/4 = 5% per quarter; 9 months = 3 quarters
2. 10000 × 1.05 × 1.05 × 1.05 = 11576.25
3. CI = 11576.25 − 10000

Answer: ₹1,576.25

📝 Exam-Level Example

Q. Find the amount on ₹16,000 at 10% per annum compounded half-yearly for 18 months.

1. Rate = 5% per half-year; 18 months = 3 periods
2. 16000 × (1.05)³ = 16000 × 1.157625

Answer: ₹18,522

🪄 Memory Trick

Convert everything to per-period language first: (rate per period, number of periods) — then it is just the normal CI formula.

⚠️ Common Mistakes

• ❌ Halving the time instead of doubling it for half-yearly compounding
• ❌ Using the yearly rate with half-yearly periods
• ❌ Counting 9 months as 2 quarters instead of 3

🏆 Exam Tips

• ✅ Memorise (1.05)² = 1.1025 and (1.05)³ = 1.157625 — half-yearly favourites
• ✅ Fractional years like 1.5 or 0.75 almost always signal a compounding-frequency question

📌 Summary

• Half-yearly → r/2, 2n; quarterly → r/4, 4n
• Work in periods, not years
• 18 months half-yearly = 3 periods
• More compounding → more interest