anthony bought a boat 3 years ago, and at that time he put a down payment of $25000 cash. today he made the second and final payment of $28 300, which includes the interest on the balance owing. anthony financed his purchase at 6.35%/a, compounded quarterly. determine the purchase price of the boat.
To determine the purchase price of the boat, we need to calculate the original balance owing before the second and final payment.
First, we need to calculate the interest rate per quarter. Since the annual interest rate is 6.35%, the quarterly interest rate will be 6.35% divided by 4 (as there are 4 quarters in a year) = 1.5875%.
Next, we need to find the number of quarters. Since Anthony bought the boat 3 years ago and interest is compounded quarterly, we multiply 3 years by 4 quarters/year = 12 quarters.
Now we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (including interest)
P = the initial principal (balance owing before final payment)
r = the interest rate per compounding period (in decimal form)
n = the number of compounding periods per year
t = the number of years
We want to find the initial principal P, so we rearrange the formula:
P = A / (1 + r/n)^(nt)
Plugging in the values we have:
A = $28,300 (the final payment amount)
r = 1.5875% (quarterly interest rate in decimal form)
n = 4 (compounding periods in a year)
t = 3 (number of years)
P = $28,300 / (1 + 0.015875/4)^(4*3)
Now we can calculate the value of P:
P = $28,300 / (1 + 0.00396875)^(12)