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# 7.1: 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

\begin{align}&;;I={{P}_{0}}r\\&;;A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)\\\end{align}

I is the interest

A is the end amount: principal plus interest

P0 is the principal (starting amount)

r is the interest rate (in decimal form. Example: 5% = 0.05)

### Example 1

A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn? P0 =$300                    the principal

r = 0.03                      3% rate

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 \begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align} I is the interest A is the end amount: principal plus interest P0 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 (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?

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

P0 = $1000 the principal r = 0.02 2% rate per half-year t = 8 4 years = 8 half-years I =$1000(0.02)(8) = $160. You will earn$160 interest total over the four years.

### 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.