Interests

Definitions

We've all heard of interest rates—whether on a mortgage, a credit card, or a loan. But what does it really mean?
Interest is essentially the "rent" you pay for borrowing money. It's the extra amount you pay to use someone else's money for a certain period.
Example of interest:
Imagine you borrow $\dollar 100$ today and promise to pay it back in one year. If you return exactly $\dollar 100$ after one year, there's no interest. However, the lender might want to be compensated for letting you use their money.
They may ask for a percentage as interest. For example, at a $10\pourcent$ interest rate per year, the interest paid is:$$\begin{aligned}\text{Interest Paid} &= \text{Percentage of the Original Amount} \\ &= \text{Interest Rate} \times \text{Original Amount} \\ &= 10\pourcent \times 100 \\ &= \frac{10}{100} \times 100 \\ &= 10~\text{dollars}\end{aligned}$$Therefore, after one year, you owe:$$\begin{aligned}\text{Amount at Year 1} &= \text{Original Amount} + \text{Interest Paid} \\ &= 100 + 10 \\ &= 110~\text{dollars}\end{aligned}$$So you would pay back $\dollar 110$ instead of $\dollar 100$. The extra $\dollar 10$ is the interest—the cost of borrowing for a year.

Definition Principal

The principal is the original amount of money that is either invested or loaned.

Definition Interest

Interest is the cost paid for borrowing money or the amount earned from lending or investing money.

Simple Interest

Discover

Suppose you borrow $\dollar 100$ with an interest rate of $10\pourcent$ per year. With simple interest, the interest is calculated only on the initial amount each year.

$ \begin{aligned}[t] \text{Total interest after 1 year} &= 10\pourcent \times 100 \\ &= \frac{10}{100} \times 100 \\ &= 10~\text{dollars} \end{aligned} $
$ \begin{aligned}[t] \text{Total interest after 2 years} &= 2 \times 10\pourcent \times 100 \\ &= 2 \times \frac{10}{100} \times 100 \\ &= 20~\text{dollars} \end{aligned} $
$ \begin{aligned}[t] \text{Total interest after 3 years} &= 3 \times 10\pourcent \times 100 \\ &= 3 \times \frac{10}{100} \times 100 \\ &= 30~\text{dollars} \end{aligned} $

These observations lead to the simple interest formula:$$\text{Simple Interest} = \text{Number of years} \times \text{Interest rate} \times \text{Principal}$$

Definition Simple Interest

The simple interest is calculated each year as a fixed percentage of the principal (original amount) borrowed or invested.

Proposition Simple Interest Formula

The simple interest, denoted by $I$, is calculated as:$$I = t \times r \times P$$where:

$P$ is the principal (original amount)
$r$ is the interest rate per year (expressed as a decimal)
$t$ is the time (in years)

The final amount, denoted by $A$, is:$$A = P + I$$

Example

Find the simple interest on a principal of $\dollar 500$ at a rate of $3\pourcent$ per year over $5$ years.

Answer

$$\begin{aligned}[t]\text{Interest} &= 5 \times 3\pourcent \times 500 \\ &= 5 \times \frac{3}{100} \times 500 \\ &= 75~\text{dollars}\end{aligned}$$