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

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:

$C \times \frac{1-(1+r)^{-time} }{rate} = PV$

C $ 500

time      48 ( 4 years x 12 months per year)

rate 0.0075 (9% annual divide by 12 months)

$500 \times \frac{1-(1+0.0075)^{-48} }{0.0075} = PV$

PV $20,092.3909

Now we recalculate n:

$C \times \frac{1-(1+r)^{-time} }{rate} = PV$

C  $675.00

time n

rate    0.0075

PV $20,092.3900

$675 \times \frac{1-(1+0.0075)^{-n} }{0.0075} = 20092.39$

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

$(1+0.0075)^{-n}= 1-\frac{20092.39\times0.0075}{675}$

$(1+0.0075)^{-n}= 0.77675122$

We use logarithmics properties to solve for n:

$-n= \frac{log0.77675122}{log(1+0.0075)}$

n = 33.8108479