Simple Interest
Learning Outcomes
- Calculate one-time simple interest, and simple interest over time
- Determine APY given an interest scenario
- Calculate compound interest
Principal and Interest
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
[latex-display]\begin{align}&I={{P}_{0}}r\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)\\\end{align}[/latex-display]- I is the interest
- A is the end amount: principal plus interest
- [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)
- r is the interest rate (in decimal form. Example: 5% = 0.05)
Examples
A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?Answer:
[latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] = $300 | the principal |
r = 0.03 | 3% rate |
I = $300(0.03) = $9. | You will earn $9 interest. |
Exercises
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?Answer: 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.
Further explanation about solving this example can be seen here. https://youtu.be/rNOEYPCnGwgSimple Interest over Time
[latex-display]\begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align}[/latex-display]- I is the interest
- A is the end amount: principal plus interest
- [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)
- r is the interest rate in decimal form
- t is time
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. [latex-display]6\div{12}=0.5[/latex-display] A 4% annual rate paid quarterly would be divided into four 1% payments. [latex-display]4\div{4}=1[/latex-display]Example
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?Answer: Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.
[latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] = $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
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 [latex]P_0[/latex] = $500 principal r = unknown t = 1 month Using [latex]I = P_0rt[/latex], we get [latex]30 = 500·r·1[/latex]. Solving, we get r = 0.06, or 6%. Since the time was monthly, this is the monthly interest. The annual rate would be 12 times this: 72% interest.
Licenses & Attributions
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- Finance. Authored by: David Lippman. Located at: http://www.opentextbookstore.com/mathinsociety/. License: CC BY-SA: Attribution-ShareAlike.
- money-grow-interest-save-invest-1604921. Authored by: TheDigitalWay. License: CC0: No Rights Reserved.
- One time simple interest. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
- Simple interest over time. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
- Simple interest T-note example. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
- Question ID 929. Authored by: Lippman,David. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
- Question ID 72476. Authored by: Day,Alyson. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.