A trust will provide $12,000 to a county library at the beginning of each 3-month period for the next 2.5 years. If the money is worth an annual rate of 6.8%, compounded quarterly, find the amount in the trust when it begins. Round your answer to the nearest cent.
A)$246,873.08
B)$111,364.28
C)$284,945.37
D)$352,106.39
E)$85,257.67

Question

Quizplus · Accepted Answer

The problem describes an annuity where payments are made at the beginning of each period, known as an annuity due. The formula to calculate the present value of an annuity due is $PV = P \times \left[\frac{1 - (1 + r)^{-n}}{r}\right] \times (1 + r)$, where $P$ is the payment amount, $r$ is the interest rate per period, and $n$ is the total number of payments. Given that the payments are $12,000 every 3 months (quarterly) for 2.5 years (which equals 10 quarters), and the annual interest rate is 6.8% (or 0.068 annually, thus 0.068/4 = 0.017 quarterly), we can calculate the present value of the annuity due as follows:$PV = 12000 \times \left[\frac{1 - (1 + 0.017)^{-10}}{0.017}\right] \times (1 + 0.017)$Calculating this gives a present value of approximately $111,364.28, which matches option B.

A Trust Will Provide $12,000 to a County Library at the Beginning