Under the installment plan, the customer pays $24/month over 4 yr with interest charged on the balance at a rate of 16%/year compounded monthly. Find the cash price for a set of encyclopedias if it is equivalent to the price paid by a customer using the installment plan.
To find the cash price for the set of encyclopedias, we need to calculate the present value of the installment plan payments.
The installment plan payments can be represented as an annuity, where the customer pays $24 per month for 4 years. The interest is charged on the balance at a rate of 16% per year, compounded monthly.
To calculate the present value of the annuity, we can use the formula:
PV = P * (1 - (1 + r)^(-n)) / r,
where PV is the present value, P is the annuity payment, r is the interest rate per period, and n is the number of periods.
In this case, P = $24, r = 16%/12 (monthly interest rate), and n = 4 * 12 (number of months in 4 years).
Let's plug in these values and calculate the present value:
PV = $24 * (1 - (1 + 0.16/12)^(-4*12)) / (0.16/12)
Simplifying the equation:
PV = $24 * (1 - (1 + 0.0133)^(-48)) / 0.0133
Calculate the expression within the parentheses:
(1 + 0.0133)^(-48) ≈ 0.4528
Substitute this value into the equation:
PV = $24 * (1 - 0.4528) / 0.0133
Simplify the equation further:
PV = $24 * 0.5472 / 0.0133
PV ≈ $986.47
Therefore, the cash price for the set of encyclopedias, equivalent to the price paid by a customer using the installment plan, is approximately $986.47.