# 4.2: Simple Interest

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Discussing interest starts with the principal, or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: $$\ 100(0.05)=\ 5$$. The total amount you would repay would be$105, the original principal plus the interest.

## Simple One-time Interest

$I=P_{0} r \nonumber$

$A=P_{0}+I=P_{0}+P_{0} r=P_{0}(1+r) \nonumber$

where

• $$I$$ is the interest
• $$A$$ is the end amount: principal plus interest
• $$P_0$$ is the principal (starting amount)
• $$r$$ is the interest rate (in decimal form. Example: $$5\% = 0.05$$)

Each year, you would earn 5% interest: $$\ 1000(0.05)=\ 50$$ in interest. So over the course of five years, you would earn a total of $250 in interest. When the bond matures, you would receive back the$1,000 you originally paid, leaving you with a total of $1,250. We can generalize this idea of simple interest over time. ## Simple Interest over Time $$I=P_{0} r t$$ $$A=P_{0}+I=P_{0}+P_{0} r t=P_{0}(1+r t)$$ where • $$I$$ is the interest • $$A$$ is the end amount: principal plus interest • $$P_0$$ is the principal (starting amount) • $$r$$ is the interest rate in decimal form • $$t$$ is time The units of measurement (years, months, etc.) for the time should match the time period for the interest rate. ## APR – Annual Percentage Rate Interest rates are usually given as an annual percentage rate (APR) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up. For example, a $$6 \%$$ APR paid monthly would be divided into twelve $$0.5 \%$$ payments. A $$4 \%$$ annual rate paid quarterly would be divided into four $$1 \%$$ payments. ## Example 3: Treasury Notes Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a$1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?

###### Solution

Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.

$$\begin{array}{ll} P_{0}=\ 1000 & \text{the principal } \\ r=0.02 & 2 \%\text{ rate per half-year} \\ t = 8 & \text{4 years = 8 half-years} \\ I=\ 1000(0.02)(8)=\ 160. & \text{You will earn }\ 160 \text{ interest total over the four years.}\end{array}$$

## Try it Now 1

A loan company charges $30 interest for a one month loan of$500. Find the annual interest rate they are charging.

$$I=\ 30$$ of interest

$$P_{0}=\ 500$$ principal

$$r= \frac{30}{500} = 0.06$$ per month

$$(0.06)(12) = 0.72$$ per year

They are charging an annual interest rate of 72%.

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