Answer: You can afford a loan of approximately $60,400.24 with a $400 per month car payment for a 4-year loan at 6% interest.
Step-by-step explanation:
To calculate how big of a loan you can afford, you can use the formula for calculating the monthly payment on a fixed-rate loan:
Monthly Payment = (Loan Amount * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-Number of Months))
Where:
Loan Amount is the principal amount you can afford to borrow.
Monthly Interest Rate is the monthly interest rate (annual interest rate divided by 12).
Number of Months is the total number of months for the loan (4 years = 48 months).
Let's calculate the Monthly Interest Rate first:
Monthly Interest Rate = (6% annual interest rate) / 12 = 0.06 / 12 = 0.005 (or 0.5%)
Now, plug the values into the formula and solve for the Loan Amount:
$400 = (Loan Amount * 0.005) / (1 - (1 + 0.005)^(-48))
Next, simplify the equation:
1 - (1 + 0.005)^(-48) ≈ 1 - (1 + 0.005)^(-48) ≈ 1 - 0.242496999 ≈ 0.757503001
Now, solve for Loan Amount:
$400 = (Loan Amount * 0.005) / 0.757503001
Multiply both sides by 0.757503001:
$400 * 0.757503001 = Loan Amount * 0.005
$302.0012004 ≈ Loan Amount * 0.005
Now, divide both sides by 0.005 to find the Loan Amount:
Loan Amount ≈ $302.0012004 / 0.005
Loan Amount ≈ $60,400.24