Interest Formula – Simple & Compound Interest Explained

The interest formula calculates the cost of borrowing money or the return on an investment — in two forms: simple interest and compound interest.

Quick Reference — Interest Formula

Simple Interest: $SI = \dfrac{P \times R \times T}{100}$

Amount (Simple): $A = P + SI = P\left(1 + \dfrac{RT}{100}\right)$

Compound Interest: $A = P\left(1 + \dfrac{r}{n}\right)^{nt}$

Compound Interest earned: $CI = A - P$

Continuously compounded: $A = Pe^{rt}$

Type: Financial mathematics — arithmetic and exponential

Used in: Banking, investment, loans, Class 8–12 curricula, competitive exams

Simple Interest Formula

Simple interest grows linearly — the interest earned each period is the same flat amount on the original principal.

$$SI = \frac{P \times R \times T}{100}$$

The total amount after $T$ periods:

$$A = P + SI = P\left(1 + \frac{RT}{100}\right)$$

Simple interest is used in short-term loans, hire purchase agreements, and bank deposit interest for some accounts.

Compound Interest Formula

Compound interest grows exponentially — interest is calculated on the principal plus accumulated interest. This is the dominant form in real-world banking and investment.

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

The compound interest earned:

$$CI = A - P = P\left[\left(1 + \frac{r}{n}\right)^{nt} - 1\right]$$

Variable Key

Compounding Frequency

More frequent compounding = more interest earned, because interest compounds on previously earned interest sooner.

Origin of Compound Interest Formula

Compound interest was described by Jacob Bernoulli (1655–1705, Switzerland) in 1683, who studied what happens as the compounding frequency increases without limit — discovering the mathematical constant $e \approx 2.71828$ in the process. The continuously compounded formula $A = Pe^{rt}$ arises from taking $n \to \infty$ in the compound interest formula. Bernoulli's discovery of $e$ through a finance problem is one of the most celebrated unexpected appearances of a mathematical constant.

Worked Examples of Interest Formula

Example 1: Simple interest

A principal of ₹5,000 is invested at 8% per annum for 3 years. Find the simple interest and total amount.

$$SI = \frac{P \times R \times T}{100} = \frac{5000 \times 8 \times 3}{100} = \frac{120000}{100} = ₹1200$$

$$A = 5000 + 1200 = ₹6200$$

Final answer: SI = ₹1,200; Amount = ₹6,200

Example 2: Compound interest (annually)

$1,000 is invested at 10% per annum compounded annually for 3 years.

A = P\left(1 + \frac{r}{n}\right)^{nt} = 1000\left(1 + \frac{0.10}{1}\right)^{1 \times 3} = 1000 \times (1.10)^3 = 1000 \times 1.331 = $1331

CI = 1331 - 1000 = $331

Final answer: Amount = $1,331; CI =$331

Example 3: Compound interest (quarterly)

$2,000 at 12% p.a. compounded quarterly for 2 years.

A = 2000\left(1 + \frac{0.12}{4}\right)^{4 \times 2} = 2000(1.03)^8 = 2000 \times 1.2668 \approx $2533.59

Final answer: Amount ≈ $2,533.59

Simple vs Compound Interest — Key differences

Common Confusions With The Interest Formula

Simple and compound interest use different variable conventions. In the simple interest formula, $R$ is the rate as a percentage (e.g. 8); in the compound formula, $r$ is the rate as a decimal (e.g. 0.08). Mixing conventions gives a wrong answer.

The total amount $A$ includes the principal. Many exam questions ask for interest earned, not total amount. Always check: if the question asks for interest, subtract $P$ from $A$.

Compounding frequency matters significantly over long periods. The same nominal rate of 12% compounds to different amounts depending on whether $n = 1$ (annually) or $n = 12$ (monthly). Monthly compounding gives approximately 12.68% effective annual rate.