Compound Interest

Interest earns interest. Money grows by the same percentage every period, which bends the curve upward faster and faster.

How it works

Exponential functions

In one sentence

A principal $P$ at annual rate $r$ compounded $n$ times a year grows to $A = P(1 + r/n)^{nt}$ after $t$ years; as $n \to \infty$ this approaches continuous compounding $Pe^{rt}$ .

General formula

{\htmlData{sym=0}{A}} = {\htmlData{sym=1}{P}}\left(1 + \frac{\htmlData{sym=2}{{r}}}{\htmlData{sym=3}{{n}}}\right)^{{\htmlData{sym=3}{n}}{\htmlData{sym=4}{t}}}

Each period multiplies the balance by $1 + r/n$ , and there are $nt$ periods.

What the symbols mean

$A$
Final amount
What the account is worth after t years.
$P$
Principal
The money you start with.
$r$
Annual rate
The yearly interest rate as a decimal: 5% is 0.05.
$n$
Compounds per year
How many times a year interest is added: 12 for monthly.
$t$
Years
How long the money grows.

$P$ is the starting amount, $r$ the yearly rate and $n$ how many times per year interest is added.
As $n \to \infty$ the formula approaches $Pe^{rt}$ , continuous compounding (dashed).
Rule of 72: money doubles in roughly $72 / (100r)$ years.

Learn the math

Equations with More Than One VariableAlgebra · Lesson 10, Solving Equations and Inequalities Exponential FunctionsAlgebra · Lesson 39, Exponential and Logarithm Functions Solving Exponential EquationsAlgebra · Lesson 41, Exponential and Logarithm Functions Applications of Exponentials and LogarithmsAlgebra · Lesson 43, Exponential and Logarithm Functions Exponential and Logarithm FunctionsCalc I · Lesson 3, Review: The Functions of Calculus Derivatives of Exponential and Logarithm FunctionsCalc I · Lesson 19, Derivatives Geometric and Telescoping SeriesCalc II · Lesson 28, Sequences and Series

Learn the math

Equations with More Than One VariableAlgebra · Lesson 10, Solving Equations and Inequalities

Exponential FunctionsAlgebra · Lesson 39, Exponential and Logarithm Functions

Solving Exponential EquationsAlgebra · Lesson 41, Exponential and Logarithm Functions

Applications of Exponentials and LogarithmsAlgebra · Lesson 43, Exponential and Logarithm Functions

Exponential and Logarithm FunctionsCalc I · Lesson 3, Review: The Functions of Calculus

Derivatives of Exponential and Logarithm FunctionsCalc I · Lesson 19, Derivatives

Geometric and Telescoping SeriesCalc II · Lesson 28, Sequences and Series