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If a borrower can afford to make monthly principal and interest payments of 1,000 and the lender will make a 30-year loan at 5-1/2% or a 20-year loan at 4-1/2% what is the largest loan (rounded to the nearest $100) this buyer can afford?
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.