Answer:
n = 33.8108479
Explanation:
We will calculate the current principal
And then calculate the time period it takes with a higher payment of 675 dollars per month:
[tex]C \times \frac{1-(1+r)^{-time} }{rate} = PV\\[/tex]
C $ 500
time 48 ( 4 years x 12 months per year)
rate 0.0075 (9% annual divide by 12 months)
[tex]500 \times \frac{1-(1+0.0075)^{-48} }{0.0075} = PV\\[/tex]
PV $20,092.3909
Now we recalculate n:
[tex]C \times \frac{1-(1+r)^{-time} }{rate} = PV\\[/tex]
C $675.00
time n
rate 0.0075
PV $20,092.3900
[tex]675 \times \frac{1-(1+0.0075)^{-n} }{0.0075} = 20092.39\\[/tex]
from the annuity formula we solve as we can until arrive at this situation:
[tex](1+0.0075)^{-n}= 1-\frac{20092.39\times0.0075}{675}[/tex]
[tex](1+0.0075)^{-n}= 0.77675122[/tex]
We use logarithmics properties to solve for n:
[tex] -n= \frac{log0.77675122}{log(1+0.0075)}[/tex]
n = 33.8108479
