Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

9.2: Simple Interest

  • Page ID
    34226
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    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

    \[I=P_{0} r\]

    \[A=P_{0}+I=P_{0}+P_{0} r=P_{0}(1+r)\]

    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: \(5\% = 0.05\))

    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?

    Solution

    \(\begin{array}{ll} P_{0}=\$ 300 & \text{the principal } \\ r=0.03 & 3 \%\text{ rate} \\
    I=\$ 300(0.03)=\$ 9. & \text{You will earn }\$ 9 \text{ interest.}\end{array}\)

    One-time simple interest is only common for extremely short-term loans. For longer term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would be earned regularly. For example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value.

    Example 2

    Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn?

    Solution

    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.

    Answer

    \(I=\$ 30\) 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%.

     


    9.2: Simple Interest is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.