Lesson 44 of 100 | Quantitative Aptitude / Compound Interest / चक्रवृद्धि ब्याज
Annual and subannual compounding
Learning outcome
Convert a nominal annual rate into matching compounding periods and handle an incomplete period only under an explicit rule.
Concepts and assumptions
An annual nominal rate r% states the rate before allowing for within-year compounding. If the arrangement specifies m equal compounding periods per year, its period rate is (r ÷ m)%. For t years, the period count is n = m × t.
A = P × [1 + r/(100 × m)]^n, with n a whole number of completed periods. Total interest = A - P.
Use m = 1 for annual, 2 for half-yearly, 4 for quarterly, and 12 for monthly compounding. Rate and time must change together: dividing the annual rate without increasing the period count misrepresents the arrangement.
The formula repeats the same per-period accumulation. Its one-year percentage growth is generally different from the quoted nominal rate. If a rate is instead given as an effective annual growth rate, do not simply divide it by m; the conversion above assumes a nominal quote.
Use 12 months per year to count periods. Annual compounding over 15 months does not, by itself, specify what happens during the extra 3 months. A question must provide an incomplete-period rule. When it specifies simple interest for leftover months, use months/12; for leftover days, use a stipulated 365-day year. Do not invent equal 30-day calendar months.
Assume positive principal, retained interest, and no extra deposits, withdrawals, interim payments or fees. Keep intermediate balances exact. Round final amount and interest to ₹0.01 only at the end, with half a paise rounded upward; use ≈ for rounded results.
Worked examples
Example 1 — Half-yearly. ₹15,000 earns a nominal 12% annually, compounded half-yearly for 18 months. Period rate = 12 ÷ 2 = 6%; periods = 18 ÷ 6 = 3. Balances are 15000 × 1.06 = ₹15,900, then 15900 × 1.06 = ₹16,854, then 16854 × 1.06 = ₹17,865.24. Interest = 17865.24 - 15000 = ₹2,865.24.
Example 2 — Quarterly, with rounding. ₹24,000 earns nominal 12%, compounded quarterly for 9 months. Rate = 12 ÷ 4 = 3% per quarter; periods = 9 ÷ 3 = 3. Balances are 24000 × 1.03 = ₹24,720, then 24720 × 1.03 = ₹25,461.60, then 25461.60 × 1.03 = ₹26,225.448 exactly. Hence A ≈ ₹26,225.45 and interest = 26225.448 - 24000 = ₹2,225.448 ≈ ₹2,225.45.
Example 3 — Explicit leftover-period rule. ₹20,000 earns 10%, compounded annually for one year, followed by simple interest for 3 months on the year-end balance. First balance = 20000 × 1.10 = ₹22,000. Extra interest = 22000 × 10 × (3/12)/100 = ₹550. Final amount = ₹22,550; total interest = ₹2,550. This result depends on the stated extension rule.
Common mistakes
Do not use the annual rate in every quarter. A fractional year may contain whole quarterly periods, but it need not contain whole annual periods. Never silently choose how an unfinished period earns interest.
Practice questions
- Find amount and interest on ₹12,500 at nominal 8%, compounded half-yearly for one year.
- Find amount and interest on ₹18,000 at nominal 12%, compounded quarterly for 6 months.
- Find amount and interest on ₹10,000 at nominal 12%, compounded monthly for 3 months.
- ₹8,000 earns 10% annually for 18 months: compound for the first year, then use simple interest on that balance for 6 months. Find amount and interest.
Worked answers
- Rate = 4%; periods = 2. A = 12500 × 1.04^2 = ₹13,520; interest = 13520 - 12500 = ₹1,020.
- Rate = 3%; periods = 2. A = 18000 × 1.03^2 = ₹19,096.20; interest = ₹1,096.20.
- Rate = 1%; periods = 3. A = 10000 × 1.01^3 = ₹10,303.01; interest = ₹303.01.
- First balance = 8000 × 1.10 = ₹8,800. Extra interest = 8800 × 10 × (6/12)/100 = ₹440. A = ₹9,240; total interest = ₹1,240.