A homeowner planning a kitchen remodeling can afford a $200 monthly payment. How much can the homeowner borrow for 3 years at 5%, compounded monthly, and still stay within the budget?
To determine how much the homeowner can borrow for 3 years at 5% compounded monthly while staying within a $200 monthly payment budget, we can use the formula for calculating the monthly payment on a loan:
P = (A * r) / (1 - (1 + r)^(-n))
Where:
P = Monthly payment
A = Loan amount
r = Monthly interest rate
n = Number of months
In this case:
P = $200
r = 5% or 0.05 (converted to decimal)
n = 3 years * 12 months/year = 36 months
Now, we can rearrange the formula to solve for the loan amount (A):
A = (P * (1 - (1 + r)^(-n))) / r
Plugging in the values:
A = ($200 * (1 - (1 + 0.05)^(-36))) / 0.05
Simplifying the equation:
A = ($200 * (1 - (1.05)^(-36))) / 0.05
Now, we can calculate the loan amount (A):
A = ($200 * (1 - 0.659735)) / 0.05
A = ($200 * 0.340265) / 0.05
A = $68,053.00 / 0.05
A = $68,053.00
Therefore, the homeowner can borrow up to $68,053.00 for 3 years at 5% compounded monthly and still stay within the $200 monthly payment budget.