Questions: Kelly and Max are looking to buy a home together and have 60,000 saved up for a down payment. Their budget allows monthly payments of 2,250 for 25 years, and the bank has offered them an annual interest rate of 5.7%, compounded monthly. What is the most expensive home they can buy?

Solution Steps

Let \( D = 60000 \) (down payment), \( P = 2250 \) (monthly payment), \( r = \frac{5.7}{100} \) (annual interest rate), and \( n = 25 \times 12 = 300 \) (total number of payments).

Convert the annual interest rate to a monthly interest rate: \[ r_m = \frac{r}{12} = \frac{0.057}{12} = 0.00475 \]

Use the present value of an ordinary annuity formula: \[ PV = P \times \left( \frac{1 - (1 + r_m)^{-n}}{r_m} \right) \] Substituting the values: \[ PV = 2250 \times \left( \frac{1 - (1 + 0.00475)^{-300}}{0.00475} \right) \]

The maximum home price \( H \) is the sum of the present value of the annuity and the down payment: \[ H = PV + D \] Substituting the calculated present value: \[ H = PV + 60000 \]

Final Answer