The formula
A = P (1 + R ÷ 100)n, where n is the number of compounding periods and R the rate per period. Compound interest is A − P. For half-yearly compounding, use R ÷ 2 per period and 2n periods; for quarterly, R ÷ 4 and 4n.
Worked examples
Example 1. CI on 10,000 at 10% for 2 years: A = 10,000 × 1.1 × 1.1 = 12,100, so CI = 2,100.
Example 2. The difference between CI and SI for 2 years at R% is P × (R ÷ 100)². On 8,000 at 5%: 8,000 × 0.0025 = 20. Check: SI = 800; CI = 8,000 × 1.05² − 8,000 = 8,820 − 8,000 = 820. Difference 20.
Example 3. 5,000 at 8% a year compounded half-yearly for 1 year: rate 4% per half-year, 2 periods: 5,000 × 1.04 × 1.04 = 5,408. CI = 408.
Example 4. A machine worth 50,000 depreciates 10% a year. Value after 2 years: 50,000 × 0.9 × 0.9 = 40,500.
Example 5. In how many years will a sum triple at 20% compounded annually? 1.2n = 3. 1.2² = 1.44; 1.2³ = 1.728; 1.2⁶ = 2.986; 1.2⁷ = 3.58. So about 6 years (and a little more than 6).
Common traps
- Using the annual rate for half-yearly periods without halving it.
- Forgetting to subtract the principal when asked for interest.
- Using the 2-year CI−SI shortcut for three years. For 3 years the difference is P(R ÷ 100)²(3 + R ÷ 100).
- Mixing growth and decay: depreciation uses (1 − R ÷ 100).
Speed tips
- For 2 years, A = P × (1 + r)². Memorise 1.05² = 1.1025, 1.1² = 1.21 and 1.2² = 1.44.
- Use CI − SI = P r² for a quick two-year check.
- For small rates and short times, approximate with SI plus a small correction.
A practice routine
- Day 1: annual compounding for 2 and 3 years, eight questions.
- Day 2: half-yearly and quarterly compounding.
- Day 3: CI versus SI differences; growth and depreciation.
- Day 4: a timed set of 12 mixed questions at 60 seconds each.
Growth patterns worth recognising
Compound growth is exponential, so small rate differences matter over time. At 10% a year, money roughly doubles in about 7 years (1.1⁷ ≈ 1.95); at 7% it takes about 10 years (1.07¹⁰ ≈ 1.97). A rough rule of thumb, the "rule of 72", says doubling time in years is about 72 divided by the rate in percent: 72 ÷ 10 = 7.2. Exams rarely ask for it directly, but it helps you sanity-check an answer: if a question says a sum doubles in 3 years at 10% compound interest, your instinct should flag it as wrong.
Quick check for any answer: compound interest for 2 years at R% should be a little more than twice the simple interest for one year. At 10% on 10,000, one year's simple interest is 1,000, twice that is 2,000 and the actual CI is 2,100: a little more, as expected.
Where it appears
Compound interest appears in SSC CGL, banking, CAT and MAT papers. Revise simple interest and percentage first if needed.
Practice tests below are published by institutes and educators on TestMint.
