Question

A sailboat costs $26,232. You pay 10% down and amortize the rest with equal monthly payments over a 8-year period. If you must pay 7.5% compounded monthly.what is your monthly payment? How much interest will you pay?Monthly payments: $(Round to two decimal places.)Interest: $(Round to two decimal places.)

Answer 1

Step 1- Write out the Present Value of Annuity formula:

$PV=P\times\frac{1-(1+\frac{r}{n})^{-t\times n}}{\frac{r}{n}}$

Where

$\begin{gathered} PV=\text{ the present value} \\ P=\text{ the periodic payment} \\ n=\text{ the number of payments in a year} \\ r=\text{ annual interest rate} \end{gathered}$

Step 2- Write out the given values and substitute them into the formula:

$\begin{gathered} PV=\ 26232(1-0.1)=\ 26232\times0.9=\ 23608.80 \\ n=12 \\ r=0.075 \end{gathered}$

Substituting the values into the formula, we have:

$23608.80=P\times\frac{1-(1+\frac{0.075}{12})^{-8\times12}}{\frac{0.075}{12}}$

Therefore,

$23608.80=72.0260P$

Dividing both sides by 72.0260, we have:

$P=\ 327.78$

Hence, the monthly payment is $327.78.

The total amount T paid is given by:

$T=\ 327.78\times8\times12=\ 31466.88$

Hence, the interest I is given by:

$I=31466.88-23608.80=\ 7858.08$

Therefore, the monthly payment is $327.78 and the interest paid is $7858.08