Interests

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 additional amount you pay to use someone else's money for a certain period of time.
Example of interest:
Imagine you borrow \(\dollar 100\) from someone today and promise to pay it back in one year. If you return exactly \(\dollar 100\) after one year, there's no interest involved. However, the lender might ask for more in return—they might want to be compensated for letting you use their money.
They may request a percentage of the amount. For example, at a \(10\pourcent\) interest rate per year, the interest you would pay is: $$ \begin{aligned} \text{Interest Paid} &= \text{Percentage of Original Amount}\\ &= \text{Interest Rate} \times \text{Original Amount}\\ &= 10\pourcent \times 100\\ &= \frac{10}{100} \times 100 \\ &= 10\text{ dollars} \end{aligned} $$Therefore, after one year, you would owe: $$ \begin{aligned} \text{Amount at Year 1} &= \text{Original Amount} + \text{Interest Paid}\\ &= 100 + 10\\ &= 110\text{ dollars} \end{aligned} $$In this case, you would pay back \(\dollar 110\) instead of \(\dollar 100\). The extra \(\dollar 10\) is the interest, which represents the cost of borrowing the money 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.

• \( \begin{aligned}[t] \text{Total interest paid after 1 year} &= \text{Percentage of Original Amount}\\&= \text{Interest Rate} \times \text{Original Amount}\\&= 10\pourcent \times 100\\&= \frac{10}{100} \times 100 \\&= 10\text{ dollars} \end{aligned} \)
• \( \begin{aligned}[t] \text{Total interest paid after 2 years} &= 2 \times \text{Percentage of Original Amount}\\&= 2 \times \text{Interest Rate} \times \text{Original Amount}\\&= 2 \times 10\pourcent \times 100\\&= 2 \times \frac{10}{100} \times 100 \\&= 20\text{ dollars} \end{aligned} \)
• \( \begin{aligned}[t] \text{Total interest paid after 3 years} &= 3 \times \text{Percentage of Original Amount}\\&= 3 \times \text{Interest Rate} \times \text{Original Amount}\\&= 3 \times 10\pourcent \times 100\\&= 3 \times \frac{10}{100} \times 100 \\&= 30\text{ dollars} \end{aligned} \)

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

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