Karl can afford car payments of $235 a month for 48 months. The bank will lend him money to buy a car at 7.75 percent interest. How much money can he afford to borrow?
PV = 235 (1 - 1.006458333..^-48)/.006458333
= $ 9672.48
To determine how much money Karl can afford to borrow, we need to calculate the loan amount that corresponds to his monthly payments.
Step 1: Convert the interest rate to a monthly rate.
To convert from an annual interest rate to a monthly interest rate, divide the annual rate by 12.
Monthly interest rate = (7.75% / 100) / 12 = 0.00645833 (approx.)
Step 2: Use the loan payment formula to calculate the loan amount.
The loan payment formula is: Loan amount = Monthly payment / (Monthly interest rate * (1 + 1 / ((1 + Monthly interest rate) ^ Number of payments - 1)))
Plugging in the values:
Monthly payment = $235
Monthly interest rate = 0.00645833
Number of payments = 48
Loan amount = $235 / (0.00645833 * (1 + 1 / ((1 + 0.00645833) ^ 48 - 1)))
Calculating this equation will give us the loan amount Karl can afford.
To find out how much money Karl can afford to borrow, we need to calculate the loan amount based on the monthly payments and the interest rate.
First, let's find the monthly interest rate by dividing the annual interest rate by 12 (since there are 12 months in a year):
Monthly Interest Rate = Annual Interest Rate / 12 = 7.75% / 12 = 0.6458% (rounded to 4 decimal places)
Next, we can use the formula to calculate the loan amount:
Loan Amount = Monthly Payment / (Monthly Interest Rate * (1 + 1 / ((1 + Monthly Interest Rate)^(number of months) - 1)))
Plugging in the given values:
Monthly Payment = $235
Monthly Interest Rate = 0.006458
Number of Months = 48
Loan Amount = $235 / (0.006458 * (1 + 1 / ((1 + 0.006458)^(48) - 1)))
Now, let's solve the equation:
Loan Amount = $235 / (0.006458 * (1 + 1 / ((1.006458)^(48) - 1)))
Loan Amount ≈ $235 / (0.006458 * (1 + 1 / (1.3619 - 1)))
Loan Amount ≈ $235 / (0.006458 * (1 + 1 / 0.3619))
Loan Amount ≈ $235 / (0.006458 * 3.7625)
Loan Amount ≈ $235 / 0.024279
Loan Amount ≈ $9665.99
Therefore, Karl can afford to borrow approximately $9666.