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Compound Interest: Annual, Half-Yearly and Quarterly Formulas

A = P(1 + r/100)ⁿ • Half-yearly: r/2 and 2n • Quarterly: r/4 and 4n • 2½ years • Different rates each year • Effective rate

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Under compound interest, each period's interest is added to the principal, and the next period's interest is worked out on that larger sum. The money grows by the same factor every period instead of the same amount, so the formula uses a power.

A = P × (1 + r/100)ⁿ, and CI = A − P

Here r is the rate per compounding period and n is the number of periods. For annual compounding those are simply the yearly rate and the number of years. The key to every variation below is getting r and n to match the compounding period.

Worked example 1: annual compounding

₹12,000 at 10% for 2 years. A = 12,000 × 1.1² = ₹14,520, so CI = ₹2,520.

Worked example 2: finding the principal

A sum amounts to ₹13,310 in 3 years at 10% compounded annually. P = 13,310 ÷ 1.1³ = 13,310 ÷ 1.331 = ₹10,000.

CompoundingRate per periodNumber of periods
Annuallyrn
Half-yearlyr ÷ 22n
Quarterlyr ÷ 44n
Monthlyr ÷ 1212n

Divide the annual rate and multiply the years by the same number. That is the whole compound interest half yearly formula: A = P × (1 + r/200)²ⁿ.

Worked example 3: half-yearly for 1½ years

₹8,000 at 10% per annum compounded half-yearly for 1½ years. The rate is 5% per half year for 3 half years. A = 8,000 × 1.05³ = ₹9,261.

Worked example 4: quarterly for 9 months

₹16,000 at 20% per annum compounded quarterly for 9 months. The rate is 5% per quarter for 3 quarters. A = 16,000 × 1.05³ = ₹18,522, so CI = ₹2,522.

When the time is not a whole number of years and compounding is annual, compound for the whole years and then apply simple interest for the leftover fraction at the proportional rate. For half a year at 10%, use 5%.

Worked example 5: 2½ years

₹20,000 at 10% for 2½ years, compounded annually. A = 20,000 × 1.1² × 1.05 = ₹25,410.

When the rate is different in each year, multiply the factors one after another. The order does not matter, because multiplication can be done in any order.

Worked example 6: 10% then 20%

₹5,000 for 2 years at 10% in the first year and 20% in the second. A = 5,000 × 1.10 × 1.20 = ₹6,600.

Because compound growth multiplies by the same factor in each period, a sum that doubles in t years becomes 4 times in 2t years and 8 times in 3t years. This is different from simple interest, where the times grow as 1 : 2 : 3 for 2, 3 and 4 times.

Worked example 7: doubling to eight times

A sum doubles in 4 years at compound interest. 8 = 2³, so it becomes 8 times in 3 × 4 = 12 years.

The effective annual rate tells you what a single annual rate would give the same result as more frequent compounding. Compute one year's growth factor and subtract 1.

Worked example 8: effective rate

10% compounded half-yearly gives 1.05² = 1.1025 in a year, an effective rate of 10.25%.

The compound interest formula is not only about money. Any quantity that changes by a fixed percentage of its current value each year follows the same rule: a town's population, the number of bacteria in a culture, or the value of a machine. For growth use (1 + r/100)ⁿ; for a decrease, such as depreciation, use (1 − r/100)ⁿ.

Worked example 9: depreciation

A machine worth ₹50,000 loses 10% of its value every year. After 2 years it is worth 50,000 × 0.9² = ₹40,500, not ₹40,000, because the second year's loss is on the reduced value.

Growth works the same way. A village of 20,000 people growing at 5% a year has 20,000 × 1.05² = 22,050 people after 2 years.

  • ✘ Halving the rate for half-yearly compounding but forgetting to double the periods.
  • ✘ Reporting the amount when the question asks for the interest, or the reverse.
  • ✘ Compounding the half year in 2½ years at the full rate. Use half the rate, once.
  • ✘ Using the simple interest rule that tripling takes twice as long as doubling.

Revise simple interest problems first if the basics are shaky. The gap between the two kinds of interest has its own shortcut in the CI and SI difference formula, and the formula list is in the Quant cheat code.

Q1. What is the compound interest on ₹5,000 at 8% per annum for 2 years, compounded annually? easy

✅ Answer: (c) ₹832

💡 Explanation: A = 5,000 × 1.08² = 5,000 × 1.1664 = ₹5,832. The compound interest is 5,832 − 5,000 = ₹832.

Q2. What is the compound interest on ₹10,000 at 20% per annum for 1 year, compounded half-yearly? easy

✅ Answer: (a) ₹2,100

💡 Explanation: Half-yearly compounding halves the rate to 10% and doubles the periods to 2. A = 10,000 × 1.1² = ₹12,100, so CI = ₹2,100.

Q3. What will ₹40,000 amount to in 6 months at 20% per annum, compounded quarterly? medium

✅ Answer: (d) ₹44,100

💡 Explanation: Quarterly compounding gives a rate of 20 ÷ 4 = 5% per quarter, and 6 months is 2 quarters. A = 40,000 × 1.05² = ₹44,100.

Q4. A sum triples itself in 5 years at compound interest. In how many years will it become 9 times itself? medium

✅ Answer: (b) 10 years

💡 Explanation: Each 5-year block multiplies the sum by 3. Nine times is 3 × 3, which needs two blocks: 2 × 5 = 10 years.

Q5. What is the amount on ₹12,000 for 1½ years at 10% per annum, compounded annually? hard

✅ Answer: (c) ₹13,860

💡 Explanation: Compound for the full year, then add simple interest at half the rate for the half year: 12,000 × 1.10 × 1.05 = ₹13,860.

Q6. What is the effective annual rate of interest for 8% per annum compounded half-yearly? medium

✅ Answer: (a) 8.16%

💡 Explanation: Half-yearly compounding gives 4% per half year. One year multiplies the sum by 1.04² = 1.0816, so the effective annual rate is 8.16%.

How do I handle 2½ years when interest is compounded annually?

Compound for the 2 whole years with the full rate, then apply simple interest for the remaining half year at half the annual rate. This is the usual convention in Indian aptitude exams unless the question says otherwise.

Does more frequent compounding always give more interest?

Yes, for the same annual rate and time. Each extra compounding lets interest earn interest sooner, so quarterly beats half-yearly, which beats annual. The gain is small at low rates and short periods.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 16 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-20 against the sources listed above.

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