Question

to help open up a wine bar, maria borrowed money from a bank. she took out a personal, amortized loan for $46,500, at an interest rate of 6.1%, with monthly payments for a term of 6 years. for each part, do not round any intermediate computations and round your final answers to the nearest cent. if necessary, refer to the list of financial formulas. (a) find marias monthly payment. (b) if maria pays the monthly payment each month for the full term, find her total amount to repay the loan. (c) if maria pays the monthly payment each month for the full term, find the total amount of interest she will pay.

Explanation:

Step1: Identify the loan - related values

$P = 46500$ (loan amount), $r=0.061$ (annual interest rate), $n = 12$ (number of payments per year), $t = 6$ (number of years).

Step2: Calculate the monthly interest rate

$i=\frac{r}{n}=\frac{0.061}{12}$

Step3: Calculate the number of payments

$m = n\times t=12\times6 = 72$

Step4: Use the amortized - loan payment formula

The formula for the monthly payment $M$ of an amortized loan is $M=\frac{P\times i\times(1 + i)^m}{(1 + i)^m-1}$.
Substitute the values: $M=\frac{46500\times\frac{0.061}{12}\times(1+\frac{0.061}{12})^{72}}{(1+\frac{0.061}{12})^{72}-1}$.
First, calculate $(1+\frac{0.061}{12})^{72}\approx1.44077$.
Then, $M=\frac{46500\times\frac{0.061}{12}\times1.44077}{1.44077 - 1}=\frac{46500\times0.0050833\times1.44077}{0.44077}$.
$M=\frac{46500\times0.007324}{0.44077}=\frac{340.566}{0.44077}\approx772.67$.

Step5: Calculate the total amount repaid

The total amount repaid $A$ if the monthly payment $M$ is made for $m$ months is $A = M\times m$.
$A=772.67\times72 = 55632.24$.

Step6: Calculate the total interest paid

The total interest $I$ paid is the total amount repaid minus the principal amount.
$I=A - P$.
$I = 55632.24-46500=9132.24$.

Answer:

(a) $\$772.67$
(b) $\$55632.24$
(c) $\$9132.24$