Question

A student looking at the timeline for a student loan on page 60 of the text makes the following​ observation: The text states that the interest rate on the loan is​ 9%, but this calculation is obviously wrong. Each monthly payment is $ 127$127​, so the student will be paying back $ 127 times 12 equals $ 1 comma 524$127×12=$1,524 per year.​ Therefore, because the principal of the loan is $ 10 comma 000$10,000​, the interest rate must be

Answer 1

Answer:

• No, I do not agree because the monthly payments are calculated using 9% interest compounded montly, and the student is calculating using simple interest.

Explanation:

The question is missing the last part. Thus, I copy the complete question for fully understanding:

• A student looking at the timeline for a student loan on page 60 of the text makes the following​ observation: The text states that the interest rate on the loan is​ 9%, but this calculation is obviously wrong. Each monthly payment is  $127×12 = $1,524 per year.​ Therefore, because the principal of the loan is $10,000​, the interest rate must be $1,524/ $10,000 = 0.1524 or 15.24%.

       Briefly explain whether you agree with the student's reasoning.

Answer

The payments of a loan are calculated using compound interest. The formula to calculate the monthly payments is:

             $Payment=L\times \frac{i(1+i)^n}{(1+i)^n-1}$

Where:

• Payment is the monthly payment

• L is the amount of the loan (principal)

• i is the monthly interest rate (the annual interest rate divided by 12)

• n is the number of months

You can obtain the monthly payment of $127 with a an interest rate of 9%, for a $10,000 loan over 10 years, using previous formula:

          $Payment=\ 10,000\times \frac{(0.09/12)(1+0.09/12)^n}{(1+0.09/12)^{10\times 12}-1}$

         $Payment=\ 126.68\approx \ 127$

The student then will pay $127 × 12 = $1524 per year, but that cannot be used to calculate the annual interest using a simple ratio, which would be valid only for simple interest.