Jill is saving $310 every month in an investment yielding 0.6% every month.  She is planning on retiring 264 months from now.  At that point she will invest in safer assets yielding 0.1% every month and she is planning on withdrawing money for 264 months.  How much will she be able to withdraw every month?

To solve this problem, we need to calculate two main parts:

1. The amount of money Jill will have saved after 264 months of investing $310 monthly at an interest rate of 0.6% per month.
2. The amount she can withdraw monthly for 264 months from the amount saved, invested at an interest rate of 0.1% per month.

### Part 1: Calculate the Future Value of Monthly Savings

The formula to calculate the future value of a series of regular monthly savings (ordinary annuity) is given by:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
where:
- $ P $ is the monthly saving amount ($310).
- $ r $ is the monthly interest rate (0.6% or 0.006).
- $ n $ is the number of months (264).

Substitute the values:
\[ FV = 310 \times \frac{(1 + 0.006)^{264} - 1}{0.006} \]

Let's calculate this step-by-step:

1. $ (1 + 0.006)^{264} $
2. Subtract 1 from the result.
3. Divide by the interest rate (0.006).
4. Multiply by the monthly saving amount ($310).

Using these steps:

\[ (1 + 0.006)^{264} \approx 5.041 \]
\[ \frac{5.041 - 1}{0.006} \approx 673.5 \]
\[ FV \approx 310 \times 673.5 \approx 208,785 \]

### Part 2: Calculate the Monthly Withdrawal

After 264 months, Jill will invest her total savings in safer assets yielding 0.1% per month and will start withdrawing for the next 264 months.

The formula to calculate the monthly withdrawal amount (PVA, Present Value Annuity) is:
\[ PVA = PV \times \frac{r}{1 - (1 + r)^{-n}} \]
where:
- $ PV $ is the future value of her savings (208,785).
- $ r $ is the new monthly interest rate (0.1% or 0.001).
- $ n $ is the number of months she will be withdrawing money (264).

Rearrange the formula to solve for the monthly withdrawal amount ($P$):
\[ P = \frac{PV \times r}{1 - (1 + r)^{-n}} \]

Substitute the values:
\[ P = \frac{208,785 \times 0.001}{1 - (1 + 0.001)^{-264}} \]

Let's calculate this step-by-step:

1. $ (1 + 0.001)^{-264} \approx 0.764 $
2. Subtract this from 1.
3. Divide the interest rate (0.001) by the result from step 2.
4. Multiply by the total savings amount (208,785).

Using these steps:

\[ 1 - 0.764 \approx 0.236 \]
\[ \frac{0.001}{0.236} \approx 0.00424 \]
\[ P \approx 208,785 \times 0.00424 \approx 885 \]

Thus, Jill will be able to withdraw approximately $885 every month for 264 months during her retirement.

Do you want more details or have any questions? 

Here are some related questions you might find useful:
1. How does compound interest differ from simple interest?
2. What is the present value of an annuity?
3. How can inflation impact retirement savings?
4. What is the difference between an annuity and a perpetuity?
5. How do varying interest rates impact long-term savings plans?

**Tip:** Regularly reviewing and adjusting your investment strategy can help ensure it aligns with your long-term financial goals.

Learn how to calculate future savings and retirement withdrawals using compound interest and annuity formulas. This problem involves determining how much Jill can save over 264 months with monthly investments and how much she can withdraw during her retirement period. Suitable for advanced high school students.

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