Simple and Compound Interest: Learn It 1

  • Calculate simple interest and compound interest
  • Determine annual percentage yield (APY) based on given interest scenarios
  • Solve for time in compound interest calculations

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. 

principal and simple interest

Principal is the amount of money that is borrowed or invested.

 

Simple interest is the interest that is calculated only on the principal amount.

For example, if you borrowed [latex]$100[/latex] from a friend and agree to repay it with [latex]5\%[/latex] interest, then the amount of interest you would pay would just be:

[latex]5\%[/latex] of [latex]100[/latex]: [latex]$100(0.05) = $5[/latex]

 

The total amount you would repay would be [latex]$105[/latex], the original principal plus the interest.

To convert a percent to a decimal, remove the [latex]\%[/latex] symbol and move the decimal place two places to the left. Ex. [latex]5\% = 0.05[/latex],  [latex]25\% = 0.25[/latex], and [latex]100\% = 1.0[/latex]. To take [latex]5\%[/latex] of [latex]$100[/latex] as in the paragraph above, write the percent as a decimal and translate the word of as multiplication.

Example: [latex]5\%[/latex] of [latex]$100 \Rightarrow 0.5\cdot100=5[/latex].

Simple One-Time Interest

Simple one-time interest refers to the interest that is earned or paid on a principal amount over a single period of time. This means that interest is only calculated once, at the end of the specified period, and it is based on the initial principal amount. Simple one-time interest is typically used in situations where the length of the investment or loan period is short and the interest rate is fixed.

simple one-time interest

To calculate simple one-time interest use the following equations:

 

[latex]\begin{align}&I={{P}_{0}}r\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)\\\end{align}[/latex]

 

  • [latex]I[/latex] is the interest
  • [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)
  • [latex]r[/latex] is the interest rate (in decimal form. Example: [latex]5\% = 0.05[/latex])
  • [latex]A[/latex] is the end amount: principal plus interest
A friend asks to borrow [latex]$300[/latex] and agrees to repay it in [latex]30[/latex] days with [latex]3\%[/latex] interest. How much interest will you earn?

Simple Interest Over Time

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 accrued (gathered) regularly.

For example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bondholder. 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.

We can generalize this idea of simple interest over time.

simple interest over time

[latex]\begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align}[/latex]

 

  • [latex]I[/latex] is the interest
  • [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)
  • [latex]r[/latex] is the interest rate in decimal form
  • [latex]t[/latex] is time
  • [latex]A[/latex] is the end amount: principal plus interest

The units of measurement (years, months, etc.) for the time should match the time period for the interest rate.

Simple Interest

 

You can view the transcript for “How to Calculate Simple Interest” here (opens in new window).

You can view the transcript for “GCSE Maths – How to Calculate Simple Interest #95” here (opens in new window).

You can view the transcript for “Simple Interest Formula” here (opens in new window).

You have probably heard the terms APR and APY before, but do you know what they mean? APR (Annual Percentage Rate) and APY (Annual Percentage Yield) are both ways to measure the interest rate on a loan or investment. APR is typically used to describe the interest rate on a loan or credit card. While, APY is used to describe the interest rate on an investment, such as a savings account or CD.

APR versus APY

APR – Annual Percentage Rate:  APR is for interest paid by consumers on loans.

 

APY – Annual Percentage Yield: APY is for interest paid to consumers on savings.

Interest rates are usually given as an annual percentage yield (APY) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APY will be divided up.

For example, a [latex]6\%[/latex] APY paid monthly would be divided into twelve [latex]0.5\%[/latex] payments.

[latex]6\div{12}=0.5[/latex]

A [latex]4\%[/latex] annual rate paid quarterly would be divided into four [latex]1\%[/latex] payments.

[latex]4\div{4}=1[/latex]

Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a [latex]$1,000[/latex] T-note with a [latex]4\%[/latex] annual rate, paid semi-annually, with a maturity in [latex]4[/latex] years. How much interest will you earn?

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