Principal (P) is the sum lent or invested, R is the rate per cent per year, T (or n) is the time in years, and Amount (A) = Principal + Interest. In simple interest the interest is worked out on the original principal every year; in compound interest each year's interest is added to the principal and itself earns interest.
Simple Interest
Simple interest
SI = P × R × T / 100; A = P + SI = P(1 + RT/100)
Rearranged: P = 100 × SI / (R × T), R = 100 × SI / (P × T), T = 100 × SI / (P × R). Rs 8000 at 12% for 3 years: SI = 8000 × 12 × 3 / 100 = Rs 2880.
- SI is the same every year, so the amount grows by equal steps. If a sum amounts to A₁ after T₁ years and A₂ after T₂ years, one year's SI = (A₂ − A₁)/(T₂ − T₁). Rs 3248 in 2 years and Rs 3696 in 4 years: 448/2 = 224 per year, P = 3248 − 448 = Rs 2800, R = 8%.
- A sum becomes k times at SI when SI = (k − 1)P. It doubles in 100/R years; if it doubles in 10 years, R = 10%.
- If a sum triples in 16 years at SI, the interest of 2P takes 16 years, so 4P (five times) takes 32 years.
Compound Interest
Compound interest (yearly)
A = P(1 + R/100)ⁿ; CI = A − P
Rs 10,000 at 10% for 3 years: A = 10,000 × 1.331 = Rs 13,310 and CI = Rs 3310. Useful factors: 1.1² = 1.21, 1.1³ = 1.331, 1.05² = 1.1025, 1.2² = 1.44, 1.2³ = 1.728.
| Compounding | Rate per period | Number of periods |
|---|---|---|
| Yearly | R | n |
| Half-yearly | R/2 | 2n |
| Quarterly | R/4 | 4n |
Half-yearly example
Rs 8000 at 10% a year compounded half-yearly for 1 year: 5% for 2 periods, A = 8000 × 1.05² = 8820, CI = Rs 820 (yearly compounding would give Rs 800). The effective yearly rate is therefore 10.25%; 12% compounded quarterly is effectively 1.03⁴ − 1 = 12.55%.
- Different rates in different years: multiply the factors. Rs 5000 at 10%, 12% and 15%: 5000 × 1.10 × 1.12 × 1.15 = Rs 7084.
- Fractional years (yearly compounding): use simple interest for the fraction. 2½ years at 10%: P × 1.1² × 1.05.
- Amounts in two consecutive years: rate = (later − earlier)/earlier × 100. Rs 2916 after 2 years and Rs 3149.28 after 3 years: 233.28/2916 = 8%.
- If a sum doubles in t years at CI, it becomes 4 times in 2t, 8 times in 3t and 16 times in 4t years.
- Present value of an amount due after n years: P = A/(1 + R/100)ⁿ. Rs 12,100 due in 2 years at 10%: 12,100/1.21 = Rs 10,000.
Difference Between CI and SI
| Period | CI − SI |
|---|---|
| 1 year (yearly compounding) | 0 (both are equal) |
| 2 years | P(R/100)² |
| 3 years | P(R/100)²(3 + R/100) |
For 2 years the difference is just the interest on the first year's interest. Rs 5000 at 10%: 5000 × 0.01 = Rs 50. For 3 years at 10% on Rs 10,000: 10,000 × 0.01 × 3.1 = Rs 310.
Growth, Depreciation and Instalments
- Population growth uses the CI formula with +R; depreciation uses (1 − R/100)ⁿ. A machine of Rs 10,00,000 depreciating at 15% a year is worth 10,00,000 × 0.85³ = Rs 6,14,125 after 3 years.
- Equal yearly instalments at CI: the loan equals the sum of the present values of the instalments. Rs 2100 at 10% repaid in 2 equal yearly instalments x: x/1.1 + x/1.21 = 2100, so x = Rs 1210.
- Interest withdrawn every year stops compounding: Rs 10,000 at 10% with interest drawn each year earns only Rs 1000 a year, like SI.
Common Mistakes
Forgetting to halve the rate and double the periods for half-yearly compounding; giving the amount when the question asks for interest; using months as years (8 months = 2/3 year); and assuming a sum that doubles in t years at CI triples in 2t years (it becomes four times).