Pierce Publishing sells encyclopedias under two payment plans: cash or installment. Under the installment plan, the customer pays $23/month over 2 yr with interest charged on the balance at a rate of 15%/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. Please round the answer to the nearest cent.
Let the cash price be $x.
Under installment plan, the customer pays $23/month for 2 years, so the total amount paid is $23/month * 24 months = $552.
To find the cash price that is equivalent to the installment plan, we need to find the present value of the $552 payments, using the formula for present value of an annuity:
PV = PMT * ((1 - (1 + r)^-n) / r)
where PV is the present value, PMT is the periodic payment, r is the interest rate per period, and n is the total number of periods.
In this case, PMT = 23, r = 0.15/12 = 0.0125 (monthly interest rate), and n = 24.
Plugging these values into the formula, we get:
PV = 23 * ((1 - (1 + 0.0125)^-24) / 0.0125) = $434.56 (rounded to the nearest cent)
Therefore, the cash price for the encyclopedias is $434.56.