Interest

What it is

Interest is the price of using someone else’s money. A bank pays you interest for leaving savings with it; a lender charges you interest for a loan. Either way it is a percent of an amount, applied over time.

$$I = Prt$$

where $P$ is the principal, $r$ is the rate as a decimal, and $t$ is the time in years.

Two kinds

	Charged on	Over time
simple	the original principal, always	grows by the same amount each year
compound	whatever the balance is now	grows faster and faster

$1000 at $10\%$:

Year	Simple	Compound
$1$	$1100	$1100
$2$	$1200	$1210
$10$	$2000	$2594

They agree for the first year and separate after it, because compound interest starts earning interest on interest. Almost every real account and loan compounds; simple interest is the version to learn first.

Interest is not the balance

The formula gives the extra money, not the total.

$$\text{balance} = P + I$$

Answering with one when the question asked for the other is the most common slip on this topic. See simple interest.

The rate needs a period

A rate with no time attached means nothing. "$5\%$" is shorthand for ”$5\%$ per year” almost everywhere, which is why a time given in months has to be divided by $12$ before it goes into the formula.