Interest (SI & CI)

What compounding means

In compound interest (CI), each year's interest is added to the principal before the next year's interest is worked out. The interest earns interest, so the amount grows geometrically — by the same factor each year, not the same rupees. $A = P\left(1 + \frac{R}{100}\right)^T, \qquad CI = A - P$

Multipliers (chips) — the everyday method

Write $1 + \frac{R}{100}$ as a fraction and multiply it T times:

The net-effect percentages

Two years at R% → net factor $\left(1+\frac{R}{100}\right)^2$: 10% → 21% (not 20%); 20% → 44%; 25% → 56.25%; 5% → 10.25%. Three years at 10% → 33.1%; at 20% → 72.8%. These small tables answer "find the amount as % of P" questions instantly. A 12.5% rate is $\frac{1}{8}$: two years multiply by $\frac{81}{64}$ → amount $= \frac{81}{64}P$, CI $= \frac{17}{64}P$. Fraction chips keep everything exact — prefer them to decimals.

Fractional years and broken periods

$2\frac{1}{2}$ years at R% (annual compounding) = 2 full years by chips, then simple interest for the half year on the amount: ₹10,000 at 10% for $2\frac{1}{2}$ years → $10000 \times 1.1 \times 1.1 = 12{,}100$; half-year SI on it $= 12100 \times \frac{5}{100} = 605$ → ₹12,705. (Compounded half-yearly instead? See the half-yearly pattern — different answer, different question.)

Compounding more than once a year

• Half-yearly: rate halves, periods double: $A = P\left(1 + \frac{R}{200}\right)^{2T}$.
• Quarterly: rate quarters, periods quadruple: $A = P\left(1 + \frac{R}{400}\right)^{4T}$. ₹10,000 at 20% p.a. half-yearly for $1\frac{1}{2}$ years → 3 half-years at 10%: $10000 \times 1.1^3 = ₹13{,}310$. More frequent compounding always gives more interest than annual compounding at the same nominal rate.

Different rates in different years

Multiply the year-wise chips: $A = P\left(1+\frac{r_1}{100}\right)\left(1+\frac{r_2}{100}\right)\cdots$ ₹25,000 with 10% then 20%: $25000 \times 1.1 \times 1.2 = ₹33{,}000$. A year of gain and a year of loss simply use their own chips.

Scaling and finding P

CI is proportional to P (same rate and time): CI on ₹30,000 = 1.5 × CI on ₹20,000. From an amount: divide by the chips. ₹6,655 after 3 years at 10% → $\frac{6655}{1.331} = ₹5{,}000$.

Quick revision

• $A = P(1 + R/100)^T$; multiply chips T times.
• Net %: 2 years at 10% → 21%; at 20% → 44%; 3 years at 10% → 33.1%.
• Half-yearly: R/2, 2T. Quarterly: R/4, 4T.
• Broken period: full years by chips, then SI on the amount.
• Different yearly rates: multiply their chips.