Question

martina borrowed money from her credit union to buy a motorcycle. she took out a personal, amortized loan for $13,500, at an interest rate of 5.7%, with monthly payments for a term of 5 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 martinas monthly payment. (b) if martina pays the monthly payment each month for the full term, find her total amount to repay the loan. (c) if martina 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 $P = 13500$, the annual interest rate $r=5.7\%=0.057$, the number of years $t = 5$ years, and the number of payments per year $n=12$.

Step2: Calculate the monthly interest rate

The monthly interest rate $i=\frac{r}{n}=\frac{0.057}{12}=0.00475$.
The number of payments $m = n\times t=12\times5 = 60$.

Step3: 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: $M = 13500\times\frac{0.00475(1 + 0.00475)^{60}}{(1+0.00475)^{60}-1}$.
First, calculate $(1 + 0.00475)^{60}$. Let $x=(1 + 0.00475)^{60}$. Using the formula $a^b=e^{b\ln(a)}$, we have $\ln(x)=60\times\ln(1.00475)\approx60\times0.00474\approx0.2844$, so $x = e^{0.2844}\approx1.32877$.
Then, $M=13500\times\frac{0.00475\times1.32877}{1.32877 - 1}=13500\times\frac{0.0063126}{0.32877}\approx13500\times0.0192007\approx259.21$.

Step4: Calculate the total amount to repay

The total amount to repay $A = M\times m$. Since $M\approx259.21$ and $m = 60$, $A=259.21\times60 = 15552.60$.

Step5: Calculate the total interest paid

The total interest paid $I=A - P$. Since $A = 15552.60$ and $P = 13500$, $I=15552.60-13500=2052.60$.

Answer:

(a) $\$259.21$
(b) $\$15552.60$
(c) $\$2052.60$