Interest is the additional amount of money applied to the initial (or principal) amount. This can be expressed as either a dollar amount or as a percentage (also referred to as a percentage rate).
In the real world, interest can be calculated when you either loan or invest money into the bank. Interest can be categorized as either simple or compound interest.
Simple Interest is when the principal amount makes additional money that increases at a linear rate.
It can be expressed algebraically as:
The total amount determined within a specific span of time can be expressed algebraically as:
Calculate the final amount of a \($1000\) investment that earns \(5\)% simple interest/year over \(15\) years. How much of the final amount is interest?
First, we can use the first simple interest formula to determine the amount of interest made:
\(\textcolor{olive}{I} = \textcolor{red}{P}\textcolor{green}{r}\textcolor{blue}{t}\)
\(\textcolor{olive}{I} = (\textcolor{red}{$1000})(\textcolor{green}{0.05})(\textcolor{blue}{15})\)
\(\textcolor{olve}{I = $750}\)
We can now use the second simple interest formula to determine the final amount:
\(A = \textcolor{red}{P} + \textcolor{olive}{I}\)
\(A = \textcolor{red}{$1000} + \textcolor{olive}{$750}\)
\(A = $1750\)
Therefore, we can determine that the final amount is \($1750\) and that \($750\) of that is interest.
First, we can determine interest by subtracting the final amount (\($2700\)) from the principal amount (\($2500\)):
\(\textcolor{olive}{I} = A - \textcolor{red}{P}\)
\(\textcolor{olive}{I} = $2700 - \textcolor{red}{$2500}\)
\(\textcolor{olive}{I} = \textcolor{olive}{$200}\)
We can use the first simple interest formula to determine the length of time needed for the principal amount to grow into its final amount:
\(\textcolor{olive}{I} = \textcolor{red}{P}\textcolor{green}{r}\textcolor{blue}{t}\)
\(\textcolor{olive}{200} = (\textcolor{red}{2500})(\textcolor{green}{0.045})\textcolor{blue}{t}\)
\(112.5t = \textcolor{olive}{200}\)
\(\cfrac{\cancel{112.5}}{\cancel{112.5}}\textcolor{blue}{t} = \cfrac{\textcolor{olive}{200}}{112.5}\)
\(t = 1.8 \; [\text{years}]\)
Therefore, we can determine that it will take \(\boldsymbol{1.8 \; [\textbf{years}]}\) for the principal to grow into its final amount.
Compound Interest is when the principal amount makes additional money that increases at an exponential rate.
The total amount determined within a specific span of time can be expressed algebraically as:
\(A = \textcolor{red}{P}(1 + \textcolor{blueviolet}{i})^{\textcolor{brown}{n}}\)
The respective formulas for finding the periodic interest rate and # of compunding periods can be algebraically described as:
These formulas can help identify variables that can be used to determine the final amount.
Calculate the final amount of a \($1000\) investment that earns \(5\)% compound interest/year over \(15\) years. How much of the final amount is interest?
First, we can determine the periodic interest rate:
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{r}}{\textcolor{magenta}{c}}\)
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{0.05}}{\textcolor{magenta}{1}}\)
\(\textcolor{blueviolet}{i} = \textcolor{blueviolet}{0.05}\)%
Using the periodic interest rate, we can now find the final amount:
\(A = \textcolor{red}{P}(1 + \textcolor{blueviolet}{i})^{\textcolor{brown}{n}}\)
\(A = \textcolor{red}{$1000}(1 + \textcolor{blueviolet}{0.05})^{\textcolor{brown}{15}}\)
\(A = \textcolor{red}{$1000}(1.05)^{\textcolor{brown}{15}}\)
\(A = \textcolor{red}{$1000}(2.08)\)
\(A = $2078.93\)
Finally, we can subtract the principal from the final amount to determine the interest made:
\(\textcolor{olive}{I} = A - \textcolor{red}{P}\)
\(\textcolor{olive}{I} = $2078.93 - \textcolor{red}{$1000}\)
\(\textcolor{olive}{I} = \textcolor{olive}{$1078.93}\)
Therefore, we can determine that the final amount is \(\boldsymbol{$2078.93}\) and \(\boldsymbol{$1078.93}\) of that is interest.
First, we can determine the # of of compounding periods:
\(\textcolor{brown}{n} = \textcolor{magenta}{c}\textcolor{blue}{t}\)
\(\textcolor{brown}{n} = (\textcolor{magenta}{12})(\textcolor{blue}{3})\)
\(\textcolor{brown}{n} = \textcolor{brown}{36}\)
Next, we can determine the periodic interest rate:
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{r}}{\textcolor{magenta}{c}}\)
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{0.0245}}{\textcolor{magenta}{12}}\)
\(\textcolor{blueviolet}{i} = \textcolor{blueviolet}{0.00204}\)%
Now we can plug all pertinent values into the amount formula, then work backwards to determine the principal amount:
\(A = \textcolor{red}{P}(1 + \textcolor{blueviolet}{i})^{\textcolor{brown}{n}}\)
\(3500 = \textcolor{red}{P}(1 + \textcolor{blueviolet}{0.00204})^{\textcolor{brown}{36}}\)
\(3500 = \textcolor{red}{P}(\textcolor{blueviolet}{1.002041667})^{\textcolor{brown}{36}}\)
\(1.076187896\textcolor{red}{P} = 3500\)
\(\cfrac{\cancel{1.076187896}\textcolor{red}{P}}{\cancel{1.076187896}} = \cfrac{3500}{1.076187896}\)
\(\textcolor{red}{P} = \textcolor{red}{$3252.22}\)
Finally, we can determine the interest made by subtracting the principal from the final amount:
\(\textcolor{olive}{I} = A - \textcolor{red}{P}\)
\(\textcolor{olive}{I} = $3500 - \textcolor{red}{$3252.22}\)
\(I = $247.78\)
Therefore, we can determine that \(\boldsymbol{$247.78}\) in interest was made over \(3\) years.
Enter in values for the fields in the following statement. Doing so will determine the final amounts for Simple Interest and Compound Interest. It will also generate a graph comparing the 2.