Question 1172845: Ron Sample is the grand prize winner in a college tuition essay contest awarded through a local organization's scholarship fund. The winner receives $3,000 at the beginning of each year for the next 5 years. How much (in $) should be invested at 8% interest compounded annually to award the prize? (Round your answer to the nearest cent.)
I did 3000 x 1-(1+.08)-5/.08 and got $11978.13
the answer seems pretty high..
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem. We need to find the present value of an annuity, where Ron Sample receives $3,000 at the beginning of each year for 5 years, and the interest rate is 8% compounded annually.
Since the payments are made at the beginning of each year, this is an annuity due.
**Formula for Present Value of an Annuity Due:**
PV = PMT * [(1 - (1 + r)^-n) / r] * (1 + r)
Where:
* PV = Present Value
* PMT = Payment per period ($3,000)
* r = Interest rate per period (8% or 0.08)
* n = Number of periods (5 years)
**Calculation:**
PV = 3000 * [(1 - (1 + 0.08)^-5) / 0.08] * (1 + 0.08)
1. Calculate (1 + 0.08)^-5:
* (1.08)^-5 ≈ 0.6805832
2. Calculate 1 - (1 + 0.08)^-5:
* 1 - 0.6805832 ≈ 0.3194168
3. Divide by 0.08:
* 0.3194168 / 0.08 ≈ 3.99271
4. Multiply by 1.08:
* 3.99271 * 1.08 ≈ 4.3121268
5. Multiply by 3000:
* 4.3121268 * 3000 ≈ 12936.3804
6. Round to the nearest cent.
* 12936.38
Therefore, approximately $12,936.38 should be invested at 8% interest compounded annually to award the prize.
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