Laurie and Craig have purchased a new home. They paid $25,000 down and have a 15-year mortgage for which they pay $980 per month. If the interest is 8.25% compounded monthly, what did the house that Laurie and Craig purchased originally cost?
To find out the original cost of the house, we need to calculate the mortgage amount, which is the total amount borrowed.
Step 1: Calculate the mortgage amount:
We know that Laurie and Craig paid a down payment of $25,000 and have a 15-year mortgage with monthly payments of $980.
Let's assume the original cost of the house is 'X'.
Down payment = $25,000
Mortgage term = 15 years = 15 * 12 = 180 months
Monthly payment = $980
The mortgage amount can be calculated using the formula for the present value of an ordinary annuity:
Mortgage amount = Monthly payment * [(1 - (1 + r)^(-n)) / r]
Where:
r = monthly interest rate = annual interest rate / 12 = 8.25% / 12 / 100 = 0.006875
n = total number of months = 180
Now substituting the values into the formula:
Mortgage amount = $980 * [(1 - (1 + 0.006875)^(-180)) / 0.006875]
Calculating this using a calculator or spreadsheet, we find that the mortgage amount is approximately $140,556.25.
Step 2: Calculate the original cost of the house:
The original cost of the house is the mortgage amount plus the down payment.
Original cost = Mortgage amount + Down payment
= $140,556.25 + $25,000
Calculating this sum, we find that the original cost of the house that Laurie and Craig purchased is approximately $165,556.25.
To find out the original cost of the house, we need to calculate the total amount paid over the 15-year mortgage period. The monthly payment of $980 remains the same throughout the entire mortgage term.
First, note that we need to convert the annual interest rate to a monthly interest rate. The formula for calculating the monthly interest rate is:
Monthly interest rate = (1 + Annual interest rate)^(1/12) - 1
Let's calculate the monthly interest rate:
Annual interest rate = 8.25% = 0.0825
Monthly interest rate = (1 + 0.0825)^(1/12) - 1
= 0.00684375
Next, we calculate the total number of monthly payments over the 15-year mortgage term:
Years = 15
Months = Years * 12
= 15 * 12
= 180
Now, let's find the future value of the mortgage payments:
FV = Monthly payment * [(1 + Monthly interest rate)^Total number of months - 1] / Monthly interest rate
FV = $980 * [(1 + 0.00684375)^180 - 1] / 0.00684375
Using a financial calculator or spreadsheet, the future value (FV) is approximately $251,425.82.
Finally, we can determine the original cost of the house by adding the down payment to the future value:
Original cost = Down payment + Future value
= $25,000 + $251,425.82
= $276,425.82
Therefore, the house that Laurie and Craig purchased originally cost approximately $276,425.82.