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Study Guides > Mathematics for the Liberal Arts

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. four rolled-up dollar bills seeming to grow out of dirt, with a miniature rake lying in between them

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.

The following video works through this example in detail. https://youtu.be/TJYq7XGB8EY  
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.

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/rNOEYPCnGwg
We can generalize this idea of simple interest over time.

Simple 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
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. [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.

This video explains the solution. https://youtu.be/IfVn20go7-Y

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

CC licensed content, Original

CC licensed content, Shared previously

  • 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.