Simple Interest

Section 2.1 Simple Interest

To determine the amount of interest one should pay for the use of someone else's money, one model is to base the amount of interest according to two conditions:

interest should be directly proportional to the length of time over which the money is lent
interest should be directly proportional to some rate which is based upon the borrower's perceived ability to repay and based upon current market conditions.

Often, when dealing with monetary value over time one can more easily illustrate the setting using a time line

\begin{equation*} \begin{matrix} \text{Money Amounts} & P & \rightarrow & A \\ \hline \\ \text{Time} & 0 & \rightarrow & t \end{matrix} \end{equation*}

Definition 2.1.1. Simple Interest.

Simple interest is a cost of funds directly proportional to the principal P, to a constant rate \(i\text{,}\) and to the length of the loan \(t\text{.}\)

Theorem 2.1.2. Simple Interest Formula.

Using the definition of simple interest 2.1.1 and the idea of direct proportionality between interest earned \(I\) and \(P\text{,}\) \(i\text{,}\) and \(t\) (respectively) yields

\begin{equation*} I = P \cdot i \cdot t \end{equation*}

and so

\begin{equation*} A = P + P \cdot i \cdot t \end{equation*}

\begin{equation*} A = P(1 + i \cdot t) \end{equation*}

Corollary 2.1.3. Linear Growth.

Under simple interest 2.1.1, the future value \(A\) of an investment grows linearly with respect to increasing time.

Checkpoint 2.1.4. WeBWorK - Simple Interest Future Value.

Note that if you have \(A\) but not \(P\) then simple algebra allows you to plug in what you know and solve for what you need.

Checkpoint 2.1.5. WeBWorK - Simple Interest Present Value.

Algebraic solving can be for any of the variables in the formula so long as you know the other three.

Essentials of the Mathematics of Finance