EXPLANATION
First, we need to apply the compounding interest equation as given below:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]Where A=balance, P=principal, r=interest rate, n=number of times interest rate is compounded (in this case the interest rate is compounded annually, so n=1)
First, we need to isolate the interest rate from each bank.
Isolating r from the Compounding interst equation:
(Dividing both sides by P):
[tex]\frac{A}{P}=(1+\frac{r}{n})^{nt}[/tex]Applying the nt root to both sides:
[tex]\sqrt[nt]{\frac{A}{P}}=(1+\frac{r}{n})[/tex]Removing the parentheses:
[tex]\sqrt[nt]{\frac{A}{P}}=1+\frac{r}{n}[/tex]Subtracting -1 to both sides:
[tex]\sqrt[nt]{\frac{A}{P}}-1=\frac{r}{n}[/tex]Multiplying both sides by n. As n=1 we can desestimate this step.Additionally, as n=1 --> nt=1*t = t
[tex]\sqrt[t]{\frac{A}{P}}-1=r[/tex]Switching sides:
[tex]r=\sqrt[t]{\frac{A}{P}}-1[/tex]Now we can compute the interest rate for each bank as follows:
[tex]r_{Super\text{ Save}}=\sqrt[6]{\frac{1173.34}{1000}}-1[/tex][tex]r_{\text{Super Save}}=\sqrt[6]{1.17334}-1[/tex][tex]r_{\text{Super Save}}=1.027000479-1=0.0700047876[/tex]As r is represented in decimal form, the r_Super Save= 2.70%
Applying the same reasoning to Star Financial:
[tex]r_{\text{Star Financial}}=\sqrt[3]{\frac{2684.35}{2500}}-1[/tex][tex]r_{\text{Star Financial}}=\sqrt[3]{1.07374}-1[/tex][tex]r_{\text{Star Financial}}=1.02399942-1=0.02399942017[/tex]As r is represented in decimal form, the r_Star Financial= 2.39%
Applying the same reasoning to Better Bank:
[tex]r_{\text{Better Bank}}=\sqrt[5]{\frac{4525.63}{4000}}-1[/tex][tex]r_{\text{Better Bank}}=\sqrt[5]{1.1314075}-1[/tex][tex]r_{\text{Better }}=1.024999871-1[/tex][tex]r_{\text{Better Bank}}=0.02499987[/tex]As r is represented in decimal form, the r_Better Bank= 2.49%
Now, that we have the interest rate values, we can build a table for each year corresponding to each Bank:
In order to draw the graph and using the table, we need to compute the balance for each year applying the above equation:
[tex]\text{B}=P(1+\frac{r}{n})^{nt}[/tex]Note: (Commas and periods are reversed, Spreadsheet Configuration issue)
