Question

You have an outstanding student loan with required payments of $500 per month for the next four years. The interest rate on the loan is 9% APR​ (compounded monthly). Now that you realize your best investment is to prepay your student​ loan, you decide to prepay as much as you can each month. Looking at your​ budget, you can afford to pay an extra $ 175 a month in addition to your required monthly payments of $500​, or $675 in total each month. How long will it take you to pay off the​ loan?

Answer 1

n = 33.8108479

Step-by-step explanation:

We will calculate the current principal

And then calculate the time period it takes with a higher payment of 675 dollars per month:

`C * (1-(1+r)^(-time) )/(rate) = PV\\`

C $ 500

time 48 ( 4 years x 12 months per year)

rate 0.0075 (9% annual divide by 12 months)

`500 * (1-(1+0.0075)^(-48) )/(0.0075) = PV\\`

PV $20,092.3909

Now we recalculate n:

`C * (1-(1+r)^(-time) )/(rate) = PV\\`

C $675.00

time n

rate 0.0075

PV $20,092.3900

`675 * (1-(1+0.0075)^(-n) )/(0.0075) = 20092.39\\`

from the annuity formula we solve as we can until arrive at this situation:

`(1+0.0075)^(-n)= 1-(20092.39*0.0075)/(675)`

`(1+0.0075)^(-n)= 0.77675122`

We use logarithmics properties to solve for n:

`-n= (log0.77675122)/(log(1+0.0075))`

n = 33.8108479