Suppose you want to buy a car. The dealer offers a financing package consisting of a 3.6% APR compounded monthly for a term of four years. Suppose that you want your monthly payments to be at most $500. What is the maximum amount that you should finance? Round your answer to the nearest dollar.

Answer 1

To find the maximum amount that should be financed, we can use the formula for monthly payments:

\[ P = \frac{r(PV)}{1 - (1 + r)^{-n}} \]

Where:
- \( P \) = monthly payment
- \( r \) = monthly interest rate (APR divided by 12)
- \( PV \) = present value or loan amount
- \( n \) = number of payments

Given:
- Monthly payment \( P = $500 \)
- \( r = \frac{3.6\%}{12} = 0.003 \)
- \( n = 4 \times 12 = 48 \) months

Substitute these values into the formula:

\[ 500 = \frac{0.003(PV)}{1 - (1 + 0.003)^{-48}} \]

\[ 500 = \frac{0.003(PV)}{1 - (1.003^{-48})} \]

\[ 500 = \frac{0.003(PV)}{1 - 0.859123} \]

\[ 500 = \frac{0.003(PV)}{0.140877} \]

\[ 500 \times 0.140877 = 0.003(PV) \]

\[ PV = \frac{500 \times 0.140877}{0.003} \]

\[ PV \approx 23302 \]

Therefore, the maximum amount that should be financed is approximately $23,302.