Question

John takes out a 4 year loan for $10800 at 6% interest compounded monthly. Calculate his monthly payment.John's monthly payment will be $The answer given is: 253.64Please break down how to solve for this.

Answer 1

- 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:

`\begin{gathered} P=PMT\left[(1-\left(1+(r)/(n)\right)^(-nt))/((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}`

- The question has given us:

`\begin{gathered} r=6\%=(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}`

- Thus, we can apply the formula above to find the monthly payment (PMT). This is done below:

`\begin{gathered} P=PMT\left[(1-\left(\right.1+(r)/(n))^(-nt))/((r)/(n))\right] \\ \\ 10800=PMT\left[(1-\left(1+(0.06)/(12)\right)^(-12*4))/((0.06)/(12))\right] \\ \\ 10800=PMT\left[(1-(1.005)^(-48))/(0.005)\right] \\ \\ 10800=PMT\left[(1-0.787098411086)/(0.005)\right] \\ \\ 10800=PMT*42.5803177828 \\ \\ \text{ Divide both sides by 42.5803177828} \\ \\ \therefore PMT=(10800)/(42.5803177828) \\ \\ PMT=253.63831371786\approx\$253.64 \end{gathered}`

Final Answer

John's monthly payment is $253.64