the academy for academic excellence has just ...
the academy for academic excellence has just announced their winner for the most successful student in mathematics. this student will receive a grant for their college education as well as national recognition for academic excellence in the field of mathematics. the grant will cover $12,177.17 of the cost of school per year for 5 years. if the initial amount is invested into an account over the five years, paying an annual interest rate of 5.5%, and you are to withdraw from this account until the remaining balance after the 5 years is zero dollars, determine the initial amount of the grant. round your answer to the nearest dollar. a. $52,000 c. $60,886 b. $55,200 d. $64,235
Answer
# Explanation:
## Step1: Identify the formula
We use the present - value of an ordinary annuity formula $PV = A\times\frac{1-(1 + r)^{-n}}{r}$, where $PV$ is the present value (initial amount), $A$ is the annual payment, $r$ is the interest rate per period, and $n$ is the number of periods.
## Step2: Define the values
$A=\$12177.17$, $r = 0.055$ (5.5% annual interest rate), and $n = 5$ years.
## Step3: Substitute values into the formula
$PV=12177.17\times\frac{1-(1 + 0.055)^{-5}}{0.055}$.
First, calculate $(1 + 0.055)^{-5}=1\div(1.055)^{5}\approx1\div1.30696\approx0.76513$.
Then, $1-(1 + 0.055)^{-5}=1 - 0.76513 = 0.23487$.
$\frac{1-(1 + 0.055)^{-5}}{0.055}=\frac{0.23487}{0.055}\approx4.27036$.
$PV=12177.17\times4.27036\approx52000$.
# Answer:
a. $\$52,000$