Interests

We've all heard of interest rates—on a mortgage, a credit card, or a loan. But what does interest really mean?
Interest is essentially the “rent” you pay for borrowing money. It is the extra amount you pay to use someone else's money for a certain period of time. The percentage used to calculate this extra amount each year is called the interest rate.
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.

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:$$\text{Simple Interest} = \text{Number of years} \times \text{Interest rate} \times \text{Principal}$$

Write the rate as a decimal: \(3\pourcent = 0.03\).$$\begin{aligned}[t]I&= t \times r \times P\\ &= 5 \times 0.03 \times 500 \\ &= 5 \times 15 \\ &= 75~\text{dollars}\end{aligned}$$The final amount is:$$\begin{aligned}[t]A &= P + I\\ &= 500 + 75\\ &= 575~\text{dollars}\end{aligned}$$

If you leave money in the bank for a period of time, the interest earned is automatically added to your account. After the interest is added, it also begins to earn interest in the next time period. This process is called compound interest.
Example of compound interest:
\(\dollar 1\,000\) is placed in an account that earns \(10\pourcent\) interest per annum (p.a.), and the interest is allowed to compound over three years. This means the account is earning \(10\pourcent\) p.a. in compound interest.
We can illustrate this in a table:

Year	Amount	Interest Earned
0	\(\dollar 1\,000\)	\(10\pourcent\) of \(\dollar 1\,000 = \dollar 100\)
1	\(\dollar 1\,000 + \dollar 100 = \dollar 1\,100\)	\(10\pourcent\) of \(\dollar 1\,100 = \dollar 110\)
2	\(\dollar 1\,100 + \dollar 110 = \dollar 1\,210\)	\(10\pourcent\) of \(\dollar 1\,210 = \dollar 121\)
3	\(\dollar 1\,210 + \dollar 121 = \dollar 1\,331\)	---

After 3 years, there will be a total of \(\dollar 1\,331\) in the account, meaning you have earned \(\dollar 331\) in compound interest.
You can also calculate the final amount using a different method:

• \(\begin{aligned}[t]\text{Amount after 1 year}&= \text{Initial amount} + \text{Interest on the initial amount} \\&= \text{Initial amount} + \text{Interest Rate} \times \text{Initial amount} \\&= 1\,000 + 0.1 \times 1\,000 \\&= 1\,000 \times (1 + 0.1) \\&= 1\,000 \times 1.1\end{aligned}\)
• \(\begin{aligned}[t]\text{Amount after 2 years}&= \text{Amount after 1 year} + \text{Interest on the amount after 1 year} \\&= \text{Amount after 1 year} + \text{Interest Rate} \times \text{Amount after 1 year} \\&= 1\,000 \times 1.1 + 0.1 \times 1\,000 \times 1.1 \\&= 1\,000 \times 1.1 \times (1 + 0.1) \\&= 1\,000 \times 1.1^2\end{aligned}\)
• \(\begin{aligned}[t]\text{Amount after 3 years}&= \text{Amount after 2 years} + \text{Interest on the amount after 2 years} \\&= \text{Amount after 2 years} + \text{Interest Rate} \times \text{Amount after 2 years} \\&= 1\,000 \times 1.1^2 + 0.1 \times 1\,000 \times 1.1^2 \\&= 1\,000 \times 1.1^2 \times (1 + 0.1) \\&= 1\,000 \times 1.1^3\end{aligned}\)

These observations lead to the compound interest formula (for interest compounded once per year):$$\text{Final amount} = \text{Initial amount}\times (1 + \text{interest rate})^{\text{number of years}} ,$$where the interest rate is written as a decimal (for example, \(10\pourcent = 0.10\)).

$$\begin{aligned}[t]A &= P (1 + r)^t \\ &= 500 \times (1 + 0.03)^5 \\ &= 500 \times 1.03^5 \\ &\approx 579.64\end{aligned}$$The final amount is approximately \(\dollar579.64\).