Question

to help open up a jewelry store, karen borrowed money from her credit union. she took out a personal, amortized loan for $49,000, at an interest rate of 6.5%, with monthly payments for a term of 8 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 karens monthly payment. (b) if karen pays the monthly payment each month for the full term, find her total amount to repay the loan. (c) if karen 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

Principal amount $P = 49000$, annual interest rate $r=6.5\%=0.065$, number of years $t = 8$ years, number of payments per year $n = 12$.

Step2: Calculate the monthly interest rate

The monthly interest rate $i=\frac{r}{n}=\frac{0.065}{12}$.

Step3: Calculate the number of payments

The number of payments $m=n\times t=12\times8 = 96$.

Step4: Use the formula for the monthly payment of an amortized loan

The formula for the monthly payment $M$ of an amortized loan is $M=P\times\frac{i(1 + i)^m}{(1 + i)^m-1}$.
Substitute the values: $i=\frac{0.065}{12}$, $P = 49000$, and $m = 96$ into the formula.
\[

$$\begin{align*} M&=49000\times\frac{\frac{0.065}{12}(1+\frac{0.065}{12})^{96}}{(1+\frac{0.065}{12})^{96}-1}\\ \end{align*}$$

\]
Using a calculator, $M\approx652.77$.

Step5: Calculate the total amount repaid

The total amount repaid $A = M\times m$. Since $M\approx652.77$ and $m = 96$, $A=652.77\times96=62665.92$.

Step6: Calculate the total interest paid

The total interest paid $I=A - P$. Since $A = 62665.92$ and $P = 49000$, $I=62665.92-49000 = 13665.92$.

Answer:

(a) $\$652.77$
(b) $\$62665.92$
(c) $\$13665.92$