Final answer:
To find the cash price of the piano, we use the present value formula for an ordinary annuity. Using the given values with a monthly interest rate of 0.006193 and 60 payments, we calculate that the cash price of the piano was approximately $6,464.77.
Step-by-step explanation:
The student is asking about the cash price of a piano which was purchased using a loan paid off with monthly payments. To determine the cash price of the piano, we use the formula for the present value of an annuity because the payments are made at the end of each period (ordinary annuity). In this case, the formula we need is PV = PMT [1 - (1 + i)^(-n)]/i, where PV is present value, PMT is the monthly payment, i is the monthly interest rate, and n is the total number of payments.
The monthly interest rate (i) is the annual rate divided by 12: i = 7.43% / 12 = 0.0743 / 12 ≈ 0.006193. The total number of payments (n) for 5 years is: n = 5 years × 12 months/year = 60 months. Substitute these values into the formula:
PV = $109 [1 - (1 + 0.006193)^(-60)] / 0.006193 ≈ $109 [1 - (1 + 0.006193)^(-60)] / 0.006193 ≈ $109 [1 - 0.633967800] / 0.006193 ≈ $109 [0.3660322] / 0.006193 ≈ $6,464.77
Therefore, the cash price of the piano was approximately $6,464.77.