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 8% 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? ​(Note: Be careful not to round any intermediate steps less than six decimal​ places.)

The number of months to pay off the loan is_____. (Round to two decimal​ places.)

Answer 1

Solution

First the loan amount is to be calculated

Loan amount=Present value of the annuity or monthly payments made

Present value of annuity =Annuity payment*((1-(1/(1+i)^m))/i)

where

i-discount or intrest rate per period-8/12=0.6666667% per month

m-number of periods -48

Present value of annuity =?

Annuity payment =500

Putting values in formula

Present value of annuity =500*((1-(1/(1+.006666667)^48))/.006666667)=20480.956321 (Loan amount)

Now

He increases the annuity payment to 675

Therefore

Present value of annuity =675*((1-(1/(1+.006666667)^m))/.006666667

20480.956321=675*((1-(1/(1+.006666667)^m))/.006666667)

Solving we get

m=34.012715

Thus number of months to payoff the loan are 34.01

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