Your parents have accumulated a $150,000 nest egg. They have been planning to use this money to pay college costs to be incurred by you and your sister, Courtney. However, Courtney has decided to forgo college and start a nail salon. Your parents are giving Courtney $24,000 to help her get started, and they have decided to take year-end vacations costing $9,000 per year for the next four years. Use 8 percent as the appropriate interest rate throughout this problem. You plan to work on a master’s and perhaps a PhD. If graduate school costs $25,620 per year, approximately how long will you be able to stay in school based on these funds?

To solve this problem, we need to calculate how long the remaining funds will last for graduate school expenses, given the various deductions, expenses, and interest rate.

### Step 1: Initial amount and deductions
Your parents have an initial nest egg of **$150,000**. They are giving **$24,000** to Courtney and will spend **$9,000** per year for vacations over the next four years.

The total deductions are:
\[
\text{Total Deductions} = 24,000 + (9,000 \times 4) = 24,000 + 36,000 = 60,000
\]

So, the remaining amount after deductions is:
\[
\text{Remaining Funds} = 150,000 - 60,000 = 90,000
\]

### Step 2: Present value of vacation expenses
However, the $36,000 for vacations is spread over four years. We need to discount these expenses to their present value using the 8% interest rate.

The present value (PV) of an annuity can be calculated with the formula:
\[
PV = \frac{C \times \left(1 - (1 + r)^{-n}\right)}{r}
\]
Where:
- $ C = 9,000 $ (the annual cost)
- $ r = 0.08 $ (the interest rate)
- $ n = 4 $ (the number of years)

Let me calculate the present value of the vacation expenses.The present value of the four years of vacation expenses is approximately **$29,809.14**.

### Step 3: Adjusting the remaining funds
Now, we subtract the present value of the vacation expenses from the original nest egg after the $24,000 gift to Courtney:

\[
\text{Adjusted Remaining Funds} = 150,000 - 24,000 - 29,809.14 = 96,190.86
\]

### Step 4: Graduate school costs
The cost of graduate school is **$25,620 per year**. We need to determine how many years of graduate school you can afford with the remaining funds, accounting for the 8% interest rate. This can be modeled using the formula for the present value of an annuity, where we solve for the number of years ($n$):

\[
PV = \frac{C \times \left(1 - (1 + r)^{-n}\right)}{r}
\]

We need to solve for $n$ given:
- $ PV = 96,190.86 $
- $ C = 25,620 $
- $ r = 0.08 $

Let me calculate the number of years.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through solving the equation manually. Let me know if you'd prefer that!

Meanwhile, do you have any other questions or want to explore something else related to this problem? Here are some related questions you might find helpful:

1. How does the 8% interest rate impact future costs?
2. What if Courtney decides to attend college later? How would that change the scenario?
3. How would the outcome differ if the interest rate were higher or lower?
4. Can the present value calculation for vacations be applied in other investment scenarios?
5. What would be the financial consequences if graduate school costs increased yearly?

**Tip**: Understanding present value is crucial when comparing financial options that involve future cash flows. It helps to account for interest and time value.

Discover how to calculate the duration of graduate school funding with a $150,000 nest egg, deductions, and an 8% interest rate. This problem involves present value calculations, annuities, and financial planning concepts, making it suitable for college-level students.

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