Compound Interest Formula

Compound interest is interest calculated on both the initial principal and the accumulated interest from previous periods.

The Formula

A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt}
A=Pert(continuous compounding)A = Pe^{rt} \quad \text{(continuous compounding)}
where PP = principal, rr = annual rate, nn = compounding periods per year, tt = 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=PertA = Pe^{rt}.

Quick Example

Invest $1000 at 6% for 10 years:
- Annual compounding: A=1000(1.06)10=$1790.85A = 1000(1.06)^{10} = \$1790.85
- Monthly: A=1000(1+0.0612)120=$1819.40A = 1000\left(1 + \frac{0.06}{12}\right)^{120} = \$1819.40
- Continuous: A=1000e0.6=$1822.12A = 1000e^{0.6} = \$1822.12

Notation

PP = principal (initial amount), rr = annual interest rate (as a decimal), nn = number of compounding periods per year, tt = time in years, AA = final amount.

What This Formula Means

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

Formal View

A=P ⁣(1+rn) ⁣ntA = P\!\left(1 + \frac{r}{n}\right)^{\!nt}; continuous limit: A=Pert=lim⁡n→∞P ⁣(1+rn) ⁣ntA = Pe^{rt} = \lim_{n \to \infty} P\!\left(1 + \frac{r}{n}\right)^{\!nt}

Worked Examples

Example 1

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

Answer

A≈$5,978.09A \approx \$5{,}978.09

First step

1
Use the formula A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt} with P=5000P = 5000, r=0.06r = 0.06, n=4n = 4, t=3t = 3.

Full solution

  1. 2
    Substitute: A=5000(1+0.064)4⋅3=5000(1.015)12A = 5000\left(1 + \frac{0.06}{4}\right)^{4 \cdot 3} = 5000(1.015)^{12}.
  2. 3
    Compute (1.015)12≈1.19562(1.015)^{12} \approx 1.19562, so A≈5000×1.19562≈5978.09A \approx 5000 \times 1.19562 \approx 5978.09.
Compound interest applies the interest rate multiple times per year. The key parameters are the principal PP, annual rate rr, compounding frequency nn, and time in years tt.

Example 2

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

Example 3

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

Common Mistakes

  • Using the annual rate rr directly when compounding more than once a year - divide by nn to get the periodic rate r/nr/n and use exponent ntnt.
  • Forgetting to convert the percent to a decimal - 6%6\% enters the formula as 0.060.06, not 66.
  • Treating compound interest like simple interest by multiplying - you must raise (1+r/n)(1+r/n) to the ntnt power, not multiply principal by rate by time.

Why This Formula Matters

It is the engine behind savings, loans, and exponential growth itself — students who treat it like simple interest underestimate long-term balances dramatically, and it sets up annuities, present/future value, and the number ee. Recognizing it by "Does each period's interest get added to the balance so the next period earns on a larger amount?" — rather than by familiar numbers — is what lets a student tell it apart from simple interest and exponential growth (general) and annuities in a mixed problem set.

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(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt} gives the amount after tt years, and A=PertA = 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=PertA = Pe^{rt}.

What do the symbols mean in the Compound Interest formula?

PP = principal (initial amount), rr = annual interest rate (as a decimal), nn = number of compounding periods per year, tt = time in years, AA = final amount.

Why is the Compound Interest formula important in Math?

It is the engine behind savings, loans, and exponential growth itself — students who treat it like simple interest underestimate long-term balances dramatically, and it sets up annuities, present/future value, and the number ee. Recognizing it by "Does each period's interest get added to the balance so the next period earns on a larger amount?" — rather than by familiar numbers — is what lets a student tell it apart from simple interest and exponential growth (general) and annuities in a mixed problem set.

What do students get wrong about Compound Interest?

The procedure for compound interest is the easy part; the trap is using the annual rate rr directly when compounding more than once a year. Asking "Does each period's interest get added to the balance so the next period earns on a larger amount?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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