Compound Interest

Under simple interest the principal never grows. Under compound interest, each year's interest is added to the principal and earns interest itself.

Amount = P(1 + R/100)^n and CI = Amount − P

Four types:

1. Amount or CI for n years — apply the formula, then subtract P if the question asked for the interest.
2. Compounded half-yearly or quarterly — halve the rate and double the periods (quarterly: rate ÷ 4, periods × 4). More frequent compounding always yields slightly more.
3. Population or growth — exactly the same formula; decay uses (1 − R/100)^n.
4. Find the rate from P and A over 2 years — (1 + R/100)² = A/P, so take the square root.

The single most common mistake is reporting the amount when the question asked for the interest. Underline "interest" or "amount" in the stem; the difference between the two answers is the whole principal, and both are printed as options.

Two shortcuts worth memorising:

• For 2 years, CI − SI = P(R/100)². So the difference for ₹8,000 at 10% is 8000 × 0.01 = ₹80.
• For 2 years, the effective rate is 2R + R²/100 percent — 10% for two years is 21%, not 20%.

A = P(1 + R/100)^n · CI = A − P · half-yearly: R/2 for 2n periods · CI − SI (2 yr) = P(R/100)² · 2-year effective rate = 2R + R²/100

Find the compound interest on ₹8,000 at 10% p.a. for 2 years. → Amount = 8000 × 1.1 × 1.1 = 8000 × 1.21 = ₹9,680, so CI = 9680 − 8000 = ₹1,680. Sanity check: simple interest would be ₹1,600, and the extra ₹80 is exactly 10% of the first year's ₹800 interest — which is also what P(R/100)² = 8000 × 0.01 predicts.