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 4.4%/year compounded monthly for a 48-month loan, what is the most expensive car that Dan can afford, assuming that he will receive $6000 for his trade-in? (Round your answer to the nearest cent.)

Answer 1

Dan can afford a car worth approximately $18,112.14, considering a monthly payment of $400, a prevailing interest rate of 4.4% per year compounded monthly for a 48-month loan, and a $6,000 trade-in.

Step-by-step explanation:

Dan's affordability for a new car can be determined using the formula for the monthly payment of a loan:

`PMT= (1+r) n −1P⋅r⋅(1+r) n ​ ,`

where PMT is the monthly payment, P is the loan amount, r is the monthly interest rate, and n is the number of payments. In Dan's case, PMT = $400),
`r= 124.4%​` = 0.00367 (monthly interest rate), and n = 48 months.

Rearranging the formula to solve for (P):

`P= r⋅(1+r) n PMT⋅((1+r) n −1)​ .`

Substituting the given values:

P = \dfrac{$400 \cdot ((1 + 0.00367)^{48} - 1)}{0.00367 \cdot (1 + 0.00367)^{48}} \approx $18,112.14.

This calculation indicates that Dan can afford a car worth around $18,112.14. To find the total amount he can spend, we add the trade-in value of $6,000. Therefore, the most expensive car Dan can afford is approximately $18,112.14 + $6,000 = $24,112.14.