Compound Interest Formula

The Formula

A = P\left(1 + \frac{r}{n}\right)^{nt}
A = Pe^{rt} \quad \text{(continuous compounding)}
where P = principal, r = annual rate, n = compounding periods per year, t = years.

When to use: Simple interest pays you only on your original deposit. Compound interest pays you interest on your interest—your money earns money on the money it already earned. The more frequently you compound, the more you earn, because each tiny interest payment starts earning its own interest sooner. The ultimate limit of compounding more and more frequently is continuous compounding: A = Pe^{rt}.

Quick Example

Invest \$1000 at 6% for 10 years:
- Annual compounding: A = 1000(1.06)^{10} = \1790.85$
- Monthly: A = 1000\left(1 + \frac{0.06}{12}\right)^{120} = \1819.40$
- Continuous: A = 1000e^{0.6} = \1822.12$

Notation

P = principal (initial amount), r = annual interest rate (as a decimal), n = number of compounding periods per year, t = time in years, A = final amount.

What This Formula Means

Interest calculated on both the initial principal and the accumulated interest from previous periods. The formula A = P\left(1 + \frac{r}{n}\right)^{nt} gives the amount after t years, and A = Pe^{rt} gives the continuously compounded amount.

Simple interest pays you only on your original deposit. Compound interest pays you interest on your interest—your money earns money on the money it already earned. The more frequently you compound, the more you earn, because each tiny interest payment starts earning its own interest sooner. The ultimate limit of compounding more and more frequently is continuous compounding: A = Pe^{rt}.

Formal View

A = P\!\left(1 + \frac{r}{n}\right)^{\!nt}; continuous limit: A = Pe^{rt} = \lim_{n \to \infty} P\!\left(1 + \frac{r}{n}\right)^{\!nt}

Worked Examples

Example 1

easy
You invest \5{,}000 at 6\%$ annual interest compounded quarterly. Find the amount after 3 years.

Solution

  1. 1
    Use the formula A = P\left(1 + \frac{r}{n}\right)^{nt} with P = 5000, r = 0.06, n = 4, t = 3.
  2. 2
    Substitute: A = 5000\left(1 + \frac{0.06}{4}\right)^{4 \cdot 3} = 5000(1.015)^{12}.
  3. 3
    Compute (1.015)^{12} \approx 1.19562, so A \approx 5000 \times 1.19562 \approx 5978.09.

Answer

A \approx \$5{,}978.09
Compound interest applies the interest rate multiple times per year. The key parameters are the principal P, annual rate r, compounding frequency n, and time in years t.

Example 2

medium
How long does it take for an investment to double at 8\% annual interest compounded monthly?

Example 3

medium
Find the amount when \2{,}000 is invested at 5\%$ compounded continuously for 4 years.

Common Mistakes

  • Using r = 6 instead of r = 0.06 for a 6% rate—always convert the percentage to a decimal.
  • Confusing the number of compounding periods: monthly means n = 12, not n = 1/12. Quarterly means n = 4.
  • Thinking continuous compounding gives dramatically more than daily compounding—the difference is usually small. The real power of compound interest comes from TIME, not compounding frequency.

Why This Formula Matters

Compound interest governs savings accounts, loans, mortgages, and investments. Understanding it is critical for personal finance. Einstein (apocryphally) called it 'the eighth wonder of the world.' The continuous form Pe^{rt} connects finance to calculus and exponential growth models.

Frequently Asked Questions

What is the Compound Interest formula?

Interest calculated on both the initial principal and the accumulated interest from previous periods. The formula A = P\left(1 + \frac{r}{n}\right)^{nt} gives the amount after t years, and A = Pe^{rt} gives the continuously compounded amount.

How do you use the Compound Interest formula?

Simple interest pays you only on your original deposit. Compound interest pays you interest on your interest—your money earns money on the money it already earned. The more frequently you compound, the more you earn, because each tiny interest payment starts earning its own interest sooner. The ultimate limit of compounding more and more frequently is continuous compounding: A = Pe^{rt}.

What do the symbols mean in the Compound Interest formula?

P = principal (initial amount), r = annual interest rate (as a decimal), n = number of compounding periods per year, t = time in years, A = final amount.

Why is the Compound Interest formula important in Math?

Compound interest governs savings accounts, loans, mortgages, and investments. Understanding it is critical for personal finance. Einstein (apocryphally) called it 'the eighth wonder of the world.' The continuous form Pe^{rt} connects finance to calculus and exponential growth models.

What do students get wrong about Compound Interest?

The rate r must be a decimal, not a percentage: 6% means r = 0.06, not r = 6. Also, r and n must use the same time unit—if r is annual, n is compoundings per year.

What should I learn before the Compound Interest formula?

Before studying the Compound Interest formula, you should understand: exponential function, e.

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Exponents and Logarithms: Rules, Proofs, and Applications →
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