# Vanna has just financed the purchase of a home for \$200,000. She agreed to repay the loan by making equal monthly blended payments of \$3000 each at 4%/a, compounded monthly. How long will it take to repay the loan? How much will the final payment be? Determine how much interest she will pay for her loan. Use Microsoft Excel to graph the amortization of the loan (Hint: Graph outstanding principal versus month). How much sooner would the loan be paid if she made a 15% down payment? How much would Vanna have saved if she had obtained a loan 3%/a, compounded monthly? Write a concluding statement about the importance of interest rates and down payments when taking out loans.

Inna Shpiro | Certified Educator

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1. The monthly payment on the mortgage is determined by the formula

`M=P*(R/(1-(1+R)^(-N)))` , where P is the principal (P = \$200,000), R is the monthly rate, expressed as a decimal (R can be found as annual rate divided by 12:

`R = 0.04/12 = 1/300` ), and N is the number of months in the terms of the mortgage. In the given example, we know the monthly payment M = \$3,000, and we need to find how long it will take Vanna to pay off the loan (N).

Plugging everything given in the formula results in

`3000 = 200,000((1/300)/(1-(1+1/300)^(-N)))`

This simplifies to

`4.5 = 1/(1-(301/300)^(-N))`

From here,

`1- (301/300)^(-N) =2/9 `

`(301/300)^(-N) = 7/9`

`N=-log_(301/300) (7/9) = 75.52`

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