Question
Dan is contemplating trading in his car for a new one. He can afford a monthly payment of at most $400. If the prevailing interest rate is 3.6%/year compounded monthly for a 48-month loan, what is the most expensive car that Dan can afford, assuming that he will receive $5000 for his trade-in? (Round your answer to the nearest cent.)
Answers
GPT 3.5
To calculate the maximum price of the car Dan can afford, we need to consider the present value of the loan of $400 per month for 48 months at an interest rate of 3.6% compounded monthly.
First, calculate the present value of the annuity using the formula:
PV = PMT * (1 - (1 + r)^-n) / r
Where:
PV = Present Value
PMT = Payment per period ($400)
r = Interest rate per period (3.6%/12 months = 0.003 per month)
n = Number of periods (48 months)
PV = $400 * (1 - (1 + 0.003)^-48) / 0.003
PV = $400 * (1 - 0.726508) / 0.003
PV = $400 * 0.273492 / 0.003
PV = $36,938.08
Now, add the trade-in value of $5000 to the present value of the loan to find the maximum price of the car Dan can afford:
Maximum Price = $36,938.08 + $5000
Maximum Price = $41,938.08
Therefore, Dan can afford a car that costs at most $41,938.08.
First, calculate the present value of the annuity using the formula:
PV = PMT * (1 - (1 + r)^-n) / r
Where:
PV = Present Value
PMT = Payment per period ($400)
r = Interest rate per period (3.6%/12 months = 0.003 per month)
n = Number of periods (48 months)
PV = $400 * (1 - (1 + 0.003)^-48) / 0.003
PV = $400 * (1 - 0.726508) / 0.003
PV = $400 * 0.273492 / 0.003
PV = $36,938.08
Now, add the trade-in value of $5000 to the present value of the loan to find the maximum price of the car Dan can afford:
Maximum Price = $36,938.08 + $5000
Maximum Price = $41,938.08
Therefore, Dan can afford a car that costs at most $41,938.08.