Dream home

You finally found your dream home. It sells for $120,000 and can be purchased by paying 10% down and financing the balance at an annual rate of 9.6% compounded monthly.

a) How much are your payments if you pay monthly for 30 years?
b) Determine how much would be paid in interest.
c) Determine the payoff after 100 payments have been made.
d) Change the rate to 8.4% and the time to 15 years and calculate the payment.
e) Determine how much would be paid in interest and compare it with the previous interest. (to the nearest dollar)

Correct answer:

a =  916.0128 USD
b =  221764.5965 USD
c =   USD
d =  1057.1976 USD
e =  82295.5613 USD

Step-by-step explanation:

$$\begin{gathered} n=30\cdot 12=360 \\ r=9.6\%=\frac{9.6}{100}=0.096 \\ v=120000-120000\cdot 0.10=108000\text{USD} \\ q=1+r/12=1+\frac{12}{125}/12=\frac{126}{125}=1.008 \\ a=v\cdot \frac{q^n\cdot (q-1)}{q^n-1}=108000\cdot \frac{{\frac{126}{125}}^{360}\cdot (\frac{126}{125}-1)}{{\frac{126}{125}}^{360}-1}=916.0128\text{USD} \end{gathered}$$

$b=n\cdot a-v=360\cdot 916.0128-108000=221764.5965\text{USD}\doteq 2.218\cdot 10^5\text{USD}$

$$\begin{gathered} m=15\cdot 12=180 \\ {r}_2=8.4\%=\frac{8.4}{100}=0.084 \\ w=1+r_2/12=1+\frac{21}{250}/12=1.007 \\ d=v\cdot \frac{w^m\cdot (w-1)}{w^m-1}=108000\cdot \frac{1.007^{180}\cdot (1.007-1)}{1.007^{180}-1}=1057.1976\text{USD} \end{gathered}$$

$$\begin{gathered} e=m\cdot d-v=180\cdot 1057.1976-108000\doteq 82295.5613\text{USD} \\ b>e \end{gathered}$$

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