Final answer:
The purchase price of the house that Sam bought was $327,533.29. We calculated this by finding the present value of the total mortgage payments and adding the down payment Sam made.
Step-by-step explanation:
To find out the purchase price of the house Sam bought, we need to determine the total amount that he borrowed from the bank, then add that to his down payment of $24,000. Since Sam paid $24,000 as a down payment and the monthly mortgage payment is $1,380 with an interest rate of 3.6% compounded monthly over 30 years, we must first calculate the present value of the mortgage to represent the amount he borrowed.
Using the formula for the present value of an annuity:
PV = Pmt × { 1 - [(1 + r) ^ -n] } / r
Where:
- PV is the present value (initial loan amount)
- Pmt is the periodic payment amount ($1,380)
- r is the monthly interest rate (3.6% per annum gives us 0.003 (3.6/12%) per month)
- n is the total number of payments (30 years × 12 months/year = 360 payments)
Plugging in the values we get:
PV = $1,380 × { 1 - [(1 + 0.003) ^ -360] } / 0.003
After calculating the formula, we find the total amount borrowed. Finally, we add the $24,000 down payment. The calculation reveals the purchase price to be $327,533.29, option b.