Question 1202787: The Johnsons have accumulated a nest egg of $50,000 that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of $2600/month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed $2900. If the Johnsons decide to secure a 15-year mortgage, what is the price range of houses that they should consider when the local mortgage rate for this type of loan is 4%/year compounded monthly? (Round your answers to the nearest cent.)
least expensive $
most expensive $
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Monthly payment formula
P = (L*i)/( 1-(1+i)^(-n) )
where,
P = monthly payment
L = loan amount
i = monthly interest rate in decimal form
n = number of months
In this case
i = 0.04/12 = 0.00333333 approximately
n = 15*12 = 180 months
The value of L is unknown, and P will take on two values as mentioned later below.
For now let's solve for L
P = (L*i)/( 1-(1+i)^(-n) )
P*( 1-(1+i)^(-n) ) = L*i
P*( 1-(1+i)^(-n) )/i = L
L = P*( 1-(1+i)^(-n) )/i
This will be useful to determine the amount the Johnsons can borrow.
If the Johnsons go with the lowest monthly payment P = 2600, then it will lead to the following:
L = P*( 1-(1+i)^(-n) )/i
L = 2600*( 1-(1+0.00333333)^(-180) )/0.00333333
L = 351,499.681766083
L = 351,499.68
Add on the $50,000 down payment to arrive at $351,499.68 + $50,000 = $401,499.68 which represents the least expensive home price they can afford.
If instead they go for the highest monthly payment P = 2900, then,
L = P*( 1-(1+i)^(-n) )/i
L = 2900*( 1-(1+0.00333333)^(-180) )/0.00333333
L = 392,057.337354478
L = 392,057.34
Add on $50,000 to get: $392,057.34 + $50,000 = $442,057.34
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Answers:
least expensive: $401,499.68
most expensive: $442,057.34
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