Simple and compound interest
Also known as: simple interest, compound interest
Simple interest is calculated only on the original principal, using I = Prt. Compound interest is calculated on the principal plus all previously earned interest, using A = P(1 + r/n)^(nt), so a balance grows faster than under simple interest.
Simple interest pays a fixed amount each period based solely on the starting principal. The formula is I = Prt, where P is the principal, r is the annual rate as a decimal, and t is the time in years; the ending balance is A = P(1 + rt). Deposit $1,000 at 10 percent simple interest for 2 years and the interest is 1,000 × 0.10 × 2 = $200, for a balance of $1,200.
Compound interest instead pays interest on interest. Its formula is A = P(1 + r/n)^(nt), where n is the number of compounding periods per year. The same $1,000 at 10 percent compounded annually for 2 years grows to 1,000(1.10)² = $1,210 — the extra $10 is the second year's interest on the first year's $100. Compounding more often raises the total: quarterly compounding uses n = 4, monthly uses n = 12, and the continuous case takes the limit, giving A = Pe^(rt).
The gap between the two widens dramatically with time, because compound growth is exponential while simple growth is linear. Over a few months the difference is negligible; over decades it dominates. This is why compound interest problems are usually framed as exponential functions, with the base (1 + r/n) acting as the growth factor per period, and why the same machinery describes population growth and depreciation.
The concept spans several exams. The GRE tests simple and compound interest directly in arithmetic and algebra, often asking for the difference between the two over a short horizon. The SAT approaches it through exponential functions and growth models. The CIMA Certificate in Business Accounting builds on it in investment appraisal, where discounting reverses compounding to produce present value and net present value.
Key takeaways
- Simple interest uses I = Prt and applies only to the original principal.
- Compound interest uses A = P(1 + r/n)^(nt) and applies to principal plus accumulated interest.
- More frequent compounding produces a larger ending balance, with continuous compounding given by A = Pe^(rt).
- $1,000 at 10 percent for 2 years grows to $1,200 simple but $1,210 compounded annually.
- Simple interest grows linearly while compound interest grows exponentially, so the gap widens over long horizons.