Question

A student takes out a loan of ​$2,200 at the beginning of each semester​ (semi-annually) for 11 semesters to pay for college. The loan charges 4.6​% interest compounded semiannually. The student graduates after the 11 semesters and refinances the loan to a lower 3.9​% rate compounded monthly with monthly payments ​(made at the end of each​ month) for 120 months. Find the monthly payment and the total interest paid.

Answer 1

To find the monthly payment and total interest paid, we use compound interest and loan formulas. The loan amount is calculated at the end of 11 semesters, and the monthly payment is determined for the refinanced loan. The total interest paid is found by subtracting the initial loan amount from the total payments over 120 months.

Step-by-step explanation:

To find the monthly payment and total interest paid, we will first calculate the loan amount at the end of the 11 semesters using the compound interest formula. Then, we will calculate the monthly payment for the refinanced loan using the loan formula.

For the initial loan, the loan amount at the end of the 11 semesters is given by:

A = P(1+r/n)^(nt)

where P = $2,200, r = 0.046 (4.6%), n = 2 (compounded semiannually), and t = 11 semesters.

Once we have the loan amount, we can use the loan formula to calculate the monthly payment for the refinanced loan. The loan formula is:

M = P * (r/n) / (1 - (1 + r/n)^(-nt))

where P is the loan amount, r = 0.039 (3.9%), n = 12 (compounded monthly), and t = 120 months.

Substituting the values into the formula, we can calculate the monthly payment. Finally, to find the total interest paid, we can subtract the initial loan amount from the total payments made over 120 months.