A couple purchasing a home budget $1700 per month for their loan payment. If they have $15,000 available for a down payment and are considering a 25-year loan, how much can they spend on the home at each of the following rates? (Round your answers to the nearest cent.)
(a) 6.9% compounded monthly
$ 1
(b) 7.1% compounded monthly
A: 205,379.74
B: 94,262.53
To determine how much the couple can spend on a home at each interest rate, we'll first calculate the monthly loan payment for each interest rate. Then, we'll use that information to find the maximum loan amount they can afford.
(a) 6.9% compounded monthly:
To find the monthly loan payment, we need to use the formula for calculating the monthly payment on an amortizing loan:
M = P [i(1 + i)^n] / [(1 + i)^n - 1]
Where:
M is the monthly payment
P is the loan principal (the cost of the home minus the down payment)
i is the monthly interest rate (annual interest rate divided by 12)
n is the total number of monthly payments (number of years multiplied by 12)
In this case, the loan principal is the cost of the home minus the down payment:
Principal = Cost of home - Down payment
Principal = (x - $15,000)
Let's solve for M using the given values:
M = $1700 (monthly budget payment)
i = 6.9% / 12 (monthly interest rate)
n = 25 years * 12 (total number of monthly payments)
M = (x - $15,000) [0.069/12(1 + 0.069/12)^(25*12)] / [(1 + 0.069/12)^(25*12) - 1]
We can now solve for x (the maximum amount they can spend on the home):
$1700 = (x - $15,000) [0.069/12(1 + 0.069/12)^(25*12)] / [(1 + 0.069/12)^(25*12) - 1]
By rearranging the equation and solving for x, we'll find the maximum amount they can spend on the home at a 6.9% interest rate.
(b) 7.1% compounded monthly:
Using the same formula as above, we'll adjust the interest rate and solve for x to find the maximum amount they can spend on the home at a 7.1% interest rate.
$1700 = (x - $15,000) [0.071/12(1 + 0.071/12)^(25*12)] / [(1 + 0.071/12)^(25*12) - 1]
By rearranging the equation and solving for x, we'll find the maximum amount they can spend on the home at a 7.1% interest rate.