Question

tammy borrowed money from an online lending company to buy a fishing boat. she took out a personal, amortized loan for $15,000, at an interest rate of 5.6%, with monthly payments for a term of 4 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 tammys monthly payment. (b) if tammy pays the monthly payment each month for the full term, find her total amount to repay the loan. (c) if tammy pays the monthly payment each month for the full term, find the total amount of interest she will pay.

Explanation:

Step1: Calculate monthly interest rate

The annual interest rate $r = 5.6\%=0.056$. The monthly interest rate $i=\frac{r}{12}=\frac{0.056}{12}$. The loan amount $P = 15000$, and the number of payments $n=4\times12 = 48$.

Step2: Use the amortized - loan payment formula

The formula for the monthly payment $M$ of an amortized loan is $M=P\frac{i(1 + i)^n}{(1 + i)^n-1}$. Substitute $P = 15000$, $i=\frac{0.056}{12}$, and $n = 48$ into the formula:
\[

$$\begin{align*} M&=15000\times\frac{\frac{0.056}{12}(1+\frac{0.056}{12})^{48}}{(1+\frac{0.056}{12})^{48}-1}\\ \end{align*}$$

\]
\[

$$\begin{align*} 1+\frac{0.056}{12}&\approx1 + 0.004667=1.004667\\ (1+\frac{0.056}{12})^{48}&\approx1.004667^{48}\approx1.25439\\ \frac{0.056}{12}&\approx0.004667\\ M&=15000\times\frac{0.004667\times1.25439}{1.25439 - 1}\\ &=15000\times\frac{0.004667\times1.25439}{0.25439}\\ &=15000\times\frac{0.005854}{0.25439}\\ &\approx345.57 \end{align*}$$

\]

Step3: Calculate total amount to repay

The total amount to repay $A = M\times n$. Since $M\approx345.57$ and $n = 48$, then $A=345.57\times48=16587.36$.

Step4: Calculate total interest paid

The total interest paid $I=A - P$. Since $A = 16587.36$ and $P = 15000$, then $I=16587.36-15000 = 1587.36$.

Answer:

(a) $\$345.57$
(b) $\$16587.36$
(c) $\$1587.36$