rachel starts an ira (individual retirement account) at the age of 22 to save for retirement. she deposits…

Answer:

$1,398,857.82$

Explanation:

Step1: Calculate the number of months

Rachel starts at 22 and retires at 65. The number of years is $65 - 22=43$ years. The number of months $n = 43\times12 = 516$ months.

Step2: Calculate the monthly interest rate

The annual interest rate $r = 8%=0.08$. The monthly interest rate $i=\frac{0.08}{12}$

Step3: Use the future - value of an ordinary annuity formula

The formula for the future value of an ordinary annuity is $F = A\times\frac{(1 + i)^{n}-1}{i}$, where $A = 300$. Substitute $A = 300$, $i=\frac{0.08}{12}$, and $n = 516$ into the formula: [ \begin{align*} F&=300\times\frac{(1+\frac{0.08}{12})^{516}-1}{\frac{0.08}{12}}\ \end{align*} ] First, calculate $(1+\frac{0.08}{12})^{516}$. Let $x=\frac{0.08}{12}\approx0.00667$. Then $(1 + x)^{n}=(1.00667)^{516}$. Using a calculator, $(1.00667)^{516}\approx4.6636$ [ \begin{align*} (1.00667)^{516}-1&\approx4.6636- 1=3.6636\ \frac{(1.00667)^{516}-1}{\frac{0.08}{12}}&=\frac{3.6636}{\frac{0.08}{12}}=\frac{3.6636\times12}{0.08}\ &=\frac{43.9632}{0.08}=549.54\ F&=300\times549.54 = 1398857.82 \end{align*} ]