Joanie takes a $6,000 loan to pay for her car. The annual interest reate on the loan is 12%. She makes no payments for 4 years, but has to pay back all the money she owes at the end of 4 years. How much more money will she owe if the interest compounds quarterly than if the interest compounds annually? Express your answer as a dollar value to the nearest cent.
P1 = Po*(1+r)^n. Compounded annually.
r = 12%/100% = 0.12 = Annual % rate expressed as a decimal.
n = 1Comp./yr. * 4yrs. = 4 Compounding
periods.
P1 = 6,000*(1.12)^4 = $9441.12
P2 = Po*(1+r)^n. Compounded quarterly.
r = (12%/4)/100% = 0.03 = Quarterly %
rate.
n = 4Comp./yr. * 4yrs. = 16 Compounding
periods.
Plug the above values into the given Eq
and solve for P2.
P2-P1 =
To calculate the difference in the amount of money Joanie will owe if the interest compounds quarterly versus annually, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan including interest
P = the principal amount (initial loan amount)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
First, let's calculate the future value of the loan with the interest compounded annually:
P = $6,000 (principal amount)
r = 0.12 (annual interest rate as a decimal)
n = 1 (compounded once a year)
t = 4 (number of years)
A = $6,000(1 + 0.12/1)^(1*4)
A = $6,000(1.12)^4
A ≈ $8,506.66
Next, let's calculate the future value of the loan with the interest compounded quarterly:
P = $6,000
r = 0.12
n = 4 (compounded four times a year)
t = 4
A = $6,000(1 + 0.12/4)^(4*4)
A = $6,000(1.03)^16
A ≈ $8,623.27
To find the difference in the amount of money Joanie will owe, subtract the amount with quarterly compounding from the amount with annual compounding:
$8,623.27 - $8,506.66 ≈ $116.61
Therefore, Joanie will owe approximately $116.61 more with quarterly compounding than with annual compounding.