the academy for academic excellence has just announced their winner for the most successful student in…

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

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$