🧩 Complex Multi-Step Problems

Carry the amount forward. Treat each year as a step: multiply the current amount by the year's multiplier $\left(1+\dfrac{R}{100}\right)$, then move on. After $T$ years the amount is $A=P\left(1+\dfrac{R}{100}\right)^T$.

Mixing SI and CI. Simple interest is flat — the same interest every year, so over $T$ years it is just $R\times T$ percent of $P$. Compound interest compounds — each year's interest earns its own interest next year. Turn each side into a single net percentage, then equate. Handy net CI rates: 10% for 2 yr = $21\%$, 20% for 2 yr = $44\%$, 10% for 3 yr = $33.1\%$.

Year-on-year trick. The jump from one year's amount to the next is exactly one year's interest on the earlier amount, so $R=\dfrac{A_{n+1}-A_n}{A_n}\times100$. A clean ratio like $9261:8820=21:20$ means a 5% rate on sight.

Equal instalments. When a CI loan is cleared in equal yearly payments $x$, each instalment's present value is found by dividing it back by the multiplier, and they must add up to the loan: $P=\dfrac{x}{1+\frac{R}{100}}+\dfrac{x}{\left(1+\frac{R}{100}\right)^2}+\dots$ Solve for $x$ — never just split $P$ equally.

Golden rule: find the simplest invariant first — often the flat SI, or a tidy ratio — before reaching for powers. Many “hard” questions collapse to a one-line calculation.