To determine the largest loan this buyer can afford, we need to calculate the monthly payment for both loan options and then compare them.
Let's start with the 30-year loan at 5-1/2% interest. To calculate the monthly payment, we use the formula:
M = P * (r * (1 + r)^n) / ((1 + r)^n - 1)
where:
M = Monthly payment
P = Principal (loan amount)
r = Monthly interest rate
n = Total number of payments (number of years * 12)
For the 30-year loan at 5-1/2% interest:
P = ? (What we are trying to find)
r = 5.5% / 100 / 12 (convert interest rate to decimal and monthly)
n = 30 * 12 (30 years converted to months)
Substituting these values into the formula, we get:
1000 = P * (0.055 / 12 * (1 + 0.055 / 12)^(30 * 12)) / ((1 + 0.055 / 12)^(30 * 12) - 1)
We can rearrange this equation to solve for P:
P = 1000 / ((0.055 / 12 * (1 + 0.055 / 12)^(30 * 12)) / ((1 + 0.055 / 12)^(30 * 12) - 1))
Now, let's calculate the monthly payment for the 20-year loan at 4-1/2% interest using the same process:
For the 20-year loan at 4-1/2% interest:
P = ? (What we are trying to find)
r = 4.5% / 100 / 12 (convert interest rate to decimal and monthly)
n = 20 * 12 (20 years converted to months)
Substituting these values into the formula, we get:
1000 = P * (0.045 / 12 * (1 + 0.045 / 12)^(20 * 12)) / ((1 + 0.045 / 12)^(20 * 12) - 1)
Again, we can rearrange this equation to solve for P:
P = 1000 / ((0.045 / 12 * (1 + 0.045 / 12)^(20 * 12)) / ((1 + 0.045 / 12)^(20 * 12) - 1))
Now, we can calculate the loan amount for both options and compare them to find the largest loan the buyer can afford. Let's plug these equations into a calculator or spreadsheet to get the results.