Comparing types of interest

A savings account with simple annual interest rate of 3.92% and an initial deposit of $5,460.75 yields a final account balance of $6,370.51 after 51 months.

Part A: Using the rate and term as the simple interest account, calculate the final account balance if the initial deposit is invested for the same number of months, with interest compounded semiannually. Show all work. (3 points)

Part B: Compare that simple interest final account balance with the compound interest final account balance. (2 point)

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]

Given:

P = $5,460.75
r = 3.92% = 0.0392
n = 2 (semi-annually)
t = 51 months = 4.25 years

Substitute the given values into the compound interest formula and solve for A:

[tex]\implies A=5460.75\left(1+\dfrac{0.0392}{2}\right)^{2 \times 4.25}[/tex]

[tex]\implies A=5460.75\left(1.0196\right)^{8.5}[/tex]

[tex]\implies A=5460.75(1.17937937...)[/tex]

[tex]\implies A=6440.2959...[/tex]

Therefore, the final account balance will be $6,440.30 (nearest cent).

Part B

The difference between the two final account balances is:

[tex]\implies 6440.30-6370.51=69.79[/tex]

Therefore, the account that earns interest compounded semi-annually is $69.79 greater than the account that earns simple interest.