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

The loan amount $P = 49000$, the annual interest rate $r=6.5\%=0.065$, the number of years $t = 8$ years, and the number of payments per year $n = 12$ (monthly payments). First, find the monthly interest rate $i=\frac{r}{n}=\frac{0.065}{12}$ and the total number of payments $m=nt=12\times8 = 96$.

Step2: Use the amortized - loan payment formula

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

$$\begin{align*} i&=\frac{0.065}{12}\approx0.0054167\\ M&=\frac{49000\times0.0054167\times(1 + 0.0054167)^{96}}{(1 + 0.0054167)^{96}-1}\\ (1 + 0.0054167)^{96}&\approx1.61907\\ M&=\frac{49000\times0.0054167\times1.61907}{1.61907 - 1}\\ &=\frac{49000\times0.0054167\times1.61907}{0.61907}\\ &=\frac{49000\times0.008773}{0.61907}\\ &=\frac{429.877}{0.61907}\\ &\approx694.39 \end{align*}$$

\]

Step3: Find the total amount to repay

The total amount to repay $A$ is the monthly payment $M$ times the total number of payments $m$. So $A = M\times m=694.39\times96 = 66661.44$.

Step4: Find the total interest paid

The total interest paid $I$ is the total amount to repay $A$ minus the principal amount $P$. So $I=A - P=66661.44-49000=17661.44$.

Answer:

(a) $\$694.39$
(b) $\$66661.44$
(c) $\$17661.44$