-	Your uncle has $310,000 invested at 7.3%, and he now wants to retire. He wants to withdraw $35,000 at the end of each year, starting at the end of this year. He also wants to have $25,000 left to give you when he ceases to withdraw funds from the account. For how many years can he make the $35,000 withdrawals and still have $25,000 left in the end?

This problem involves determining the number of years your uncle can make withdrawals from an investment while leaving a specified amount at the end. This is a classic problem involving the time value of money, specifically the future value of an annuity.

### Steps to Solve:

1. **Initial Information:**
   - Principal amount: $ P = \$310,000 $
   - Annual interest rate: $ r = 7.3\% $ or 0.073
   - Annual withdrawal amount: $ W = \$35,000 $
   - Desired final amount: $ F = \$25,000 $

2. **Future Value of Annuity Formula:**
   The future value $ F $ after $ n $ years when withdrawing $ W $ at the end of each year from an account earning interest rate $ r $ is given by:

   \[
   F = P(1 + r)^n - \frac{W \times \left((1 + r)^n - 1\right)}{r}
   \]

   Where:
   - $ F $ is the amount left after the last withdrawal.
   - $ P $ is the initial investment.
   - $ r $ is the annual interest rate.
   - $ W $ is the annual withdrawal amount.
   - $ n $ is the number of years.

3. **Substituting the known values:**

   \[
   25,000 = 310,000(1 + 0.073)^n - \frac{35,000 \times \left((1 + 0.073)^n - 1\right)}{0.073}
   \]

4. **Simplifying the equation:**

   This equation is not straightforward to solve algebraically, so it typically requires numerical methods or trial and error to find $ n $. However, it can be solved using logarithms or financial calculators to find the exact number of years $ n $.

Let's compute $ n $ using a calculator or numerically:

First, compute the first term $ P(1 + r)^n $, and then adjust until the equation balances.

To make the calculation, I will solve the equation numerically.

---

Let me calculate the value of $ n $.Your uncle can make the $35,000 withdrawals for approximately **14 years** and still have $25,000 left in the account at the end.

Would you like any further details on how this was calculated, or do you have any questions? Here are some related questions for further exploration:

1. What happens to the number of years if the interest rate changes?
2. How would the result change if your uncle wanted to leave $50,000 instead of $25,000?
3. What if the withdrawals were made monthly instead of yearly?
4. How would a different initial investment amount affect the duration of the withdrawals?
5. What is the impact of inflation on these withdrawals?

**Tip:** Understanding the time value of money and how annuities work can help in planning long-term investments and withdrawals effectively.

Your uncle has $310,000 invested at 7.3%, and he now wants to retire. He wants to withdraw $35,000 at the end of each year, starting at the end of this year. He also wants to have $25,000 left to give you when he ceases to withdraw funds from the account. For how many years can he make the $35,000 withdrawals and still have $25,000 left in the end?

Determine how long your uncle can make $35,000 yearly withdrawals from a $310,000 investment at 7.3% interest and still have $25,000 left at the end. This problem explores annuities, time value of money, and future value calculations, ideal for Grades 11-12 and financial math learners.

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