Question 1131695: Carissa's parents were unable to pay for her last year of college, so she obtained a student loan of $9,000. The conditions of the loan were: She would make no payments while in college, but the interest would accumulate at 3.0% compounded monthly. Upon graduation she would begin equal monthly payments that would repay the loan in 4 years. (Round your answers to the nearest cent.)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i used the following online calculator to assist me in solving this.
https://arachnoid.com/finance/
the loan is for $9,000.
no payments are made while in college.
interest will be 3% compounded monthly.
the loan is for the last year in college.
the formula to find the future value of the loan after 1 year is:
f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period
n is then number of time periods.
the time periods are months.
in your problem:
f is what you want to find.
p is 9,000
r = 3% / 100 = .03 / 12 = .0025
n = 1 * 12 = 12
the formula becomes f = 9000 * (1 + .0025) ^ 12.
the result is f = 9273.743612
that future value becomes the present value of the loan.
the use of a financial calculator is recommended here, even though there is a formula that can be used as well.
financial formulas can be found at https://www.algebra.com/algebra/homework/Finance/THEO-2016-04-29.lesson#notes
the particular formula used here would be:
ANNUITY FOR A PRESENT AMOUNT WITH END OF TIME PERIOD PAYMENTS
a = (p*r)/(1-(1/(1+r)^n))
a is the annuity.
p is the present amount.
r is the interest rate per time period.
n is the number of time periods.
i used the calculator, but the formula should provide you with the same answer.
the main difference between the use of either is that the formula uses the interest rate while the calculator uses the interest rate percent.
rate = rate percent / 100.
rate percent = rate * 100.
the inputs to the calculator would be:
present value = 9273.743612
future value = 0
number of time periods = 12 * 4 = 48
interest rate percent per time period = 3/12 = .25
payments are made at the end of each time period.
click on PMT and you get the payments required at the end of each month.
the calculator tells you that the payment required is 205.27 at the end of each month.
the payment is shown as negative because it is money going out, while the present value is shown as positive because it is money that has already come in.
at the end of the 48 months, the remaining balance of money owed is equal to 0 and the loan is paid off.
here's what the results of the calculator look like.
in the formula above, a is the annuity which is the same as the monthly payment.
if you try to use the formula, your inputs would be:
p = 9273.73612
r = .03/12 = .0025
n = 4 * 12 = 48
the formula of a = (p*r)/(1-(1/(1+r)^n)) becomes:
a = (9273.73612 * .0025) / (1 - (1 / (1 + .0025) ^ 48 )).
the result is a = 205.2680735 which is rounded to 205.27.
when you use the formula, be sure to make your entries into your calculator exactly as shown since any deviation could result in a wrong answer.
use of the calculator is just so much easier.
if you want to know the total interest that was paid, you would multiply the loan payments * 48 and then subtract the principal of the loan.
that would be 205.2680735 * 48 - 9273.73612 = 579.1239171 rounded to 579.12.
the general concepts used here are:
the present value of the loan repayment is the future value of the loan after 1 year.
the total interest paid is the payment at the end of each month times the number of months minus the present value of the loan repayment.
for what you needed to do, the solution is:
not exactly sure, but i suspect the value of the monthly payment, which is 205.27 payable at the end of each month for a period of 48 months.
the interest rate on the loan is assumed to be the same throughout the loan period, which includes the 1 year of no payments plus 4 years of payments.
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