Answer:
Their monthly house payment is of $2,184.65.
Step-by-step explanation:
Compound interest:
The compound interest formula is given by:
![A(t) = P(1 + \frac{r}{n})^{nt}](https://tex.z-dn.net/?f=A%28t%29%20%3D%20P%281%20%2B%20%5Cfrac%7Br%7D%7Bn%7D%29%5E%7Bnt%7D)
Where A(t) is the amount of money after t years, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per year and t is the time in years for which the money is invested or borrowed.
A family buys a new home for $212,500 and pays a 20% down payment ($42,500).
This means that the loan is of 212,500 - 42,500 = $170,000, that is, ![P = 170,000](https://tex.z-dn.net/?f=P%20%3D%20170%2C000)
Value of the loan in 15 years:
15 years means that ![t = 15](https://tex.z-dn.net/?f=t%20%3D%2015)
5.75% interest means that ![r = 0.0575](https://tex.z-dn.net/?f=r%20%3D%200.0575)
Compounded yearly, so ![n = 1](https://tex.z-dn.net/?f=n%20%3D%201)
Then
![A(t) = P(1 + \frac{r}{n})^{nt}](https://tex.z-dn.net/?f=A%28t%29%20%3D%20P%281%20%2B%20%5Cfrac%7Br%7D%7Bn%7D%29%5E%7Bnt%7D)
![A(15) = 170000(1 + \frac{0.0575}{1})^{15}](https://tex.z-dn.net/?f=A%2815%29%20%3D%20170000%281%20%2B%20%5Cfrac%7B0.0575%7D%7B1%7D%29%5E%7B15%7D)
![A(15) = 393237](https://tex.z-dn.net/?f=A%2815%29%20%3D%20393237)
Monthly payment:
Total of $393,237 in 15*12 months. So
![M = \frac{393237}{15*12} = 2184.65](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7B393237%7D%7B15%2A12%7D%20%3D%202184.65)
Their monthly house payment is of $2,184.65.