(6 marks, difficulty level: Easy/Medium) You are thinking about buying a piece of art that costs $30,000. The art dealer is proposing the following deal: he will lend you the money, and you will repay the loan by making the same payment every 3 years for the next 24 years (i.e., a total of 8 payments). If the interest rate is 7% per year compounded annually, how much will you have to pay every three years?

Question

(6 marks, difficulty level: Easy/Medium) You are thinking about buying a piece of art that costs ​$30,000. The art dealer is proposing the following​ deal: he will lend you the​ money, and you will repay the loan by making the same payment every 3 years for the next 24 years​ (i.e., a total of 8 ​payments). If the interest rate is 7​% per year compounded​ annually, how much will you have to pay every three​ years?

BlackTom AI · Accepted Answer

To start, let’s identify the basic structure of the problem: we are financing a $30,000 purchase with a loan that is repaid in eight equal payments every three years over 24 years, with an annual interest rate of 7% compounded annually.
First, convert the annual rate to a rate that matches the payment interval. Since payments occur every 3 years, the effective interest rate over a 3-year period is (1 + 0.07)^3 − 1 ≈ 1.225043 − 1 = 0.225043, i ≈ 22.5043% per 3-year period.
Next, treat this as a level-payment annuity with 8 payments (n = 8) at the per-period rate i ≈ 0.225043. The present value factor for an ordinary annuity is a-angle-n = [1 − (1 + i)^(−n)] / i.
Compute (1 + i)^(−n) = (1.225043)^(−8). Estimating (1.225043)^8 ≈ 5.10 gives (1.225043)^(−8) ≈ 0.196. Then 1 − 0.196 ≈ 0.804.
Divide by i: 0.804 / 0.225043 ≈ 3.57. This is the present value factor for eight payments at the 3-year interval rate.
Finally, set the loan amount equal to the present value of the eight payments: 30,000 ≈ Payment × 3.57, so Payment ≈ 30,000 / 3.57 ≈ 8,400. The closest option is $8,409.
Now let’s evaluate the other options to pinpoint why they are not correct.
- $7,819: If you divided 30,000 by 3.57 you get about 8,400; 7,819 would imply a smaller annuity factor (~3.84), which would occur if the interest rate were lower or the number of payments were larger. Neither aligns with the stated 7% annual rate and 8 payments, so this is inconsistent with the given data.
- $9,181: This would correspond to a larger annuity factor (around 3.27). That would happen if the rate were higher or if there were fewer payments. Again, that contradicts the 7% annual rate and 8 payments over 24 years.
- $12,108: This is significantly larger and would imply an even larger present value factor or a much smaller rate. It overshoots the required present value given 8 payments over 3-year intervals at 7% annual rate, making it implausible as the fair level payment.
In summary, the calculation aligns with an 8-payment annuity at a 3-year interval with a 7% annual rate, yielding a payment close to $8,409, which matches the correct choice.

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