Compound interest questions, with answers

The questions, with answers

1. 1.What is the compound interest formula, and how is it different from simple interest?

   In short: Amount = P × (1 + R/100)^n, and compound interest is that amount minus P — interest earned on the principal plus all interest already added.

   Rs 5,000 at 8% a year compounded annually for 2 years amounts to 5000 × 1.08² = 5000 × 1.1664 = Rs 5,832, so the compound interest is Rs 832. Simple interest on the same terms is 5000 × 8 × 2 / 100 = Rs 800. The extra Rs 32 is exactly the interest on the first year's interest: Rs 400 earning 8% for one year. That is the whole difference between the two: in year one they agree, and from year two compound interest earns on a larger base. A question that asks for 'the interest' wants CI = A − P; one that says 'amounts to' wants A. Read which one it is before computing.

2. 2.How do you work out compound interest without a calculator?

   In short: Apply the rate year by year to the running total, or use the combined rate: 2R + R²/100 per cent for two years, 3R + 3R²/100 + R³/10000 for three.

   Year by year is usually fastest: Rs 6,000 at 5% becomes 6,300 after one year, 6,615 after two and 6,945.75 after three, so three years' interest is Rs 945.75. The combined-rate shortcut gets there in one multiplication: for three years at 5% it is 15 + 0.75 + 0.0125 = 15.7625%, and 15.7625% of 6,000 is 945.75. The two-year versions are worth knowing for common rates — 10% gives 21%, 20% gives 44%, 5% gives 10.25% — because they turn a squared bracket into a single percentage. Avoid long multiplication of decimals such as 1.05³; the running total, one percentage step at a time, is less error-prone.

3. 3.How do you handle half-yearly and quarterly compounding?

   In short: Divide the annual rate by the number of compounding periods in a year, multiply the years by that same number, and apply the ordinary formula.

   Half-yearly at 8% a year means 4% per half-year; quarterly at 8% means 2% per quarter. Rs 20,000 at 8% compounded half-yearly for 1½ years is three periods at 4%: 20000 × 1.04³ = 20000 × 1.124864 = Rs 22,497.28, so the interest is Rs 2,497.28. The same money compounded annually would earn a little less, because more frequent compounding adds interest on interest sooner — 8% compounded half-yearly is an effective 1.04² − 1 = 8.16% a year. The error to avoid is using the annual rate with the number of periods, or the period rate with the number of years; convert both together.

4. 4.What is the difference between compound and simple interest over two and three years?

   In short: For two years the difference is P × (R/100)²; for three years it is P × (R/100)² × (3 + R/100).

   The two-year difference is the interest on the first year's interest, which the simple interest page works through. Three years needs one more term. On Rs 15,000 at 10% for three years, the difference is 15000 × 0.01 × 3.1 = Rs 465. Check it directly: compound interest is 15000 × (1.331 − 1) = Rs 4,965, simple interest is 15000 × 0.3 = Rs 4,500, and the gap is Rs 465. The formula runs backwards too: if the three-year difference at 10% is Rs 93, then P × 0.031 = 93 and P = Rs 3,000. For two years at 12%, a difference of Rs 72 means P × 0.0144 = 72, so P = Rs 5,000.

5. 5.How do you find the principal when the amount is given?

   In short: Divide the amount by (1 + R/100)^n — never take the rate off the amount, because the amount already includes interest on interest.

   A sum amounts to Rs 8,640 in 2 years at 20% compounded annually. The growth factor is 1.2² = 1.44, so the principal is 8640 / 1.44 = Rs 6,000 and the interest was Rs 2,640. Two wrong answers are usually among the options. Taking 40% off the amount gives 8640 × 0.6 = Rs 5,184, which treats the amount as the base. Dividing by 1.4 gives about Rs 6,171, which uses simple interest's growth. Both come from forgetting that the second year's interest was earned on the first year's total. When the numbers do not divide cleanly, work backwards year by year: divide by 1.2 once for each year.

6. 6.How do you find the rate or the number of years from two amounts?

   In short: Amounts in consecutive years differ by the factor (1 + R/100), so divide one year's amount by the year before; for time, find which power of the factor fits.

   A sum amounts to Rs 6,760 after 2 years and Rs 7,030.40 after 3 years. One year's growth took 6,760 to 7,030.40, a factor of 1.04, so the rate is 4%, and the principal is 6760 / 1.04² = 6760 / 1.0816 = Rs 6,250. The difference, Rs 270.40, is also one year's interest on 6,760 — 4% of it — which is a quick check. For time, compare the amount with the principal: Rs 12,500 at 20% becomes Rs 21,600 after how many years? 21600 / 12500 = 1.728, and 1.2³ = 1.728, so 3 years. Knowing the first powers of 1.1, 1.2 and 1.25 — 1.21 and 1.331; 1.44 and 1.728; 1.5625 and 1.953125 — turns these into recognition instead of arithmetic.

7. 7.How long does a sum take to double or triple at compound interest?

   In short: If a sum doubles in n years it becomes 4 times in 2n and 8 times in 3n; the rule of 72 estimates the doubling time as 72 ÷ R years.

   Compound growth multiplies, so equal stretches of time give equal multiples. A sum that doubles in 5 years becomes 4 times in 10 and 8 times in 15; one that triples in 4 years becomes 9 times in 8 and 27 times in 12. That is the contrast with simple interest, where a sum that doubles in 5 years needs 15 years, not 10, to become 4 times. For an estimate, divide 72 by the rate: at 12% money doubles in about 6 years (1.12⁶ is 1.97, just short of 2), and at 8% in about 9 (1.08⁹ is 2.00). The rule is an approximation, so use it to eliminate options, not to write an exact answer.

8. 8.How do population growth and depreciation questions use compound interest?

   In short: Growth multiplies by (1 + r/100) every year and depreciation by (1 − r/100); when the rate changes from year to year, multiply each year's factor.

   A town of 1,20,000 people growing at 5% a year has 1,20,000 × 1.05² = 1,32,300 people after two years. A machine bought for Rs 50,000 that loses 10% of its value each year is worth 50000 × 0.9² = Rs 40,500 after two years — not Rs 40,000, because the second year's loss is 10% of the reduced value. Changing rates multiply the same way: growth of 10% followed by 20% is 1.1 × 1.2 = 1.32, a 32% rise, not 30%. Running the formula backwards gives the past: a population of 1,32,300 after two years of 5% growth was 1,32,300 / 1.1025 = 1,20,000 two years earlier.

How the diagnostic asks it

One question from the Aptitude bank, exactly as a sitting would show it. The bank has 8 on compound interest and 84 across Aptitude.