Respuesta :
Solution
- The question tells us that John takes out a loan worth $10,800. That is the present value of the loan. The interest rate is 6% and he is to repay portions of the loan monthly. These portions of the loan will add up to the amount of the loan initially acquired plus the extra 6% which makes up the $10,800.
- The formula representing this whole process is given below:
[tex]\begin{gathered} P=PMT\left[\frac{1-\left(1+\frac{r}{n}\right)^{-nt}}{\frac{r}{n}}\right] \\ \\ where, \\ P=\text{ Present value Principal} \\ PMT=\text{ Payment} \\ r=\text{ Annual percentage rate in decimal} \\ n=\text{ The number of payments made per year} \end{gathered}[/tex]- The question has given us:
[tex]\begin{gathered} r=6\%=\frac{6}{100}=0.06 \\ \\ n=\text{ 12. This is so because the interest is compounded monthly and there are 12 months} \\ \text{ in a year} \\ \\ t=4\text{ years} \\ \\ P=\$10,800 \end{gathered}[/tex]- Thus, we can apply the formula above to find the monthly payment (PMT). This is done below:
[tex]\begin{gathered} P=PMT\left[\frac{1-\left(\right.1+\frac{r}{n})^{-nt}}{\frac{r}{n}}\right] \\ \\ 10800=PMT\left[\frac{1-\left(1+\frac{0.06}{12}\right)^{-12\times4}}{\frac{0.06}{12}}\right] \\ \\ 10800=PMT\left[\frac{1-(1.005)^{-48}}{0.005}\right] \\ \\ 10800=PMT\left[\frac{1-0.787098411086}{0.005}\right] \\ \\ 10800=PMT\times42.5803177828 \\ \\ \text{ Divide both sides by 42.5803177828} \\ \\ \therefore PMT=\frac{10800}{42.5803177828} \\ \\ PMT=253.63831371786\approx\$253.64 \end{gathered}[/tex]Final Answer
John's monthly payment is $253.64
