Compound Interest • Topic 2 of 4

Half-Yearly & Quarterly

When interest is compounded more than once a year, two things change together: the rate per period shrinks and the number of periods grows. For half-yearly compounding use half the rate for twice as many periods — A = P(1 + r/200)^(2n) — and for quarterly use a quarter of the rate for four times as many periods, A = P(1 + r/400)^(4n). The classic CAT trap is to halve the rate but forget to double the time (or vice versa); always change both. A nominal 10% per annum compounded half-yearly is really 5% per half-year, which over a year gives 1.05² = 1.1025, an effective 10.25% — more than 10%, because the first half-year’s interest itself earns interest in the second half. The more frequent the compounding, the higher the effective rate, so half-yearly always beats annual at the same nominal rate.

✅ Solved examples

1. Find the amount on ₹10,000 at 10% per annum for 1 year, compounded half-yearly.
Rate per half-year = 5%, periods = 2. A = 10000 × 1.05² = 10000 × 1.1025 = ₹11,025.
2. Find the CI on ₹8,000 at 20% per annum for 1 year, compounded half-yearly.
10% per half-year, 2 periods. A = 8000 × 1.1² = 8000 × 1.21 = ₹9,680. CI = ₹1,680.
3. Find the amount on ₹16,000 at 20% per annum for 9 months, compounded quarterly.
5% per quarter, 3 quarters. A = 16000 × 1.05³ = 16000 × 1.157625 = ₹18,522.
4. What is the effective annual rate when 8% per annum is compounded quarterly?
2% per quarter, 4 periods. (1.02)⁴ = 1.08243… ⇒ effective ≈ 8.24% per annum.

✏️ Practice — try these, take hints as needed

1. Find the amount on ₹4,000 at 10% p.a. for 1 year, compounded half-yearly.
5% per half-year.
2 periods, so 1.05².
4000 × 1.1025.
₹4,410
2. Find the CI on ₹20,000 at 10% p.a. for 1½ years, compounded half-yearly.
5% per half-year.
1½ years = 3 periods.
A = 20000 × 1.05³, then subtract 20000.
₹3,152.50
3. Find the amount on ₹12,000 at 40% p.a. for 6 months, compounded quarterly.
10% per quarter.
6 months = 2 quarters.
12000 × 1.1².
₹14,520
4. Which is larger: 12% p.a. compounded annually or 12% p.a. compounded half-yearly?
Half-yearly = 6% twice.
Compare 1.12 with 1.06².
1.06² = 1.1236.
Half-yearly (effective 12.36%)
5. Find the effective annual rate when 20% p.a. is compounded half-yearly.
10% per half-year.
(1.1)² − 1.
= 0.21.
21%

📝 Topic test — 8 questions

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