Answer:
If she invests $3000 per year,
Monica will have 216157.33 at the end of 20 years
Monica will have 723998.05 at the end of 30 years
Monica will have 2301274.26 at the end of 40 years
When Monica invests $250 per month she will have
$247313.84 after 20 years
$873741.03 after 30 years
$2941193.13 after 40 years
Monica'll have to pay $882,357.94 as taxes on monthly contributions, and $690,382.28 on annual contributions.
Since Monica intends to contribute fixed amounts at regular intervals into a Roth IRA, we consider these payments as annuities.
We need to compute the Future Values (FV) the annuities. For the first set of calculations the contributions are made annually. So we use the following formula:
\mathbf{FV = A* \frac{(1+r)^{n} -1}{r}}
Substituting the values in the formula above we get,
\mathbf{FV = 3000* \frac{(1+0.12)^{n} -1}{.12}}f she
\mathbf{FV = A* \frac{(1.12)^{n} -1}{0.12}}
We will change the value of n in order to determine the amount at the end of 20, 30 and 40 years
\mathbf{FV_{20} = 3000* \frac{(1.12)^{20} -1}{0.12} = 216157.33 }
\mathbf{FV_{30} = 3000* \frac{(1.12)^{30} -1}{0.12} = 723998.05}
\mathbf{FV_{40} = 3000* \frac{(1.12)^{40} -1}{0.12} = 2301274.26}
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In the next set of calculations Monica deposits $25 each month to IRA, the FV is calculated as follows:
\mathbf{FV = A* \frac{(1+\frac{r}{m})^{mn} -1}{\frac{r}{m}}}
We plug in the values to get
\mathbf{FV_{20} = 250* \frac{(1+\frac{0.12}{12})^{12*20} -1}{\frac{0.12}{12}}}
\mathbf{FV_{20} = 250* \frac{(1.01)^{240} -1}{0.01} = 247313.84 }
\mathbf{FV_{30} = 250* \frac{(1.01)^{360} -1}{0.01} = 873741.03 }
\mathbf{FV_{40} = 250* \frac{(1.01)^{480} -1}{0.01} = 2941193.13 }
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If she has a 30% tax rate after 40 years, she'd have to pay \mathbf{0.3 * 2941193.13 = 882357.94} on monthly contributions.
If her tax rate is 30% after 40 years, she'd have to pay \mathbf{0.3 * 2301274.26 = 690382.28 } on annual contributions.