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?

We need to determine how long you can stay in school based on the remaining funds after deducting the amounts given to your sister and for the yearly vacations, taking into account the 8% interest rate. Here's how we will approach the problem:

1. **Initial Fund**: The total nest egg starts at $150,000.
2. **Deduction for Sister**: Your parents give your sister $24,000 to start her salon.
3. **Vacation Costs**: Your parents will take four vacations at $9,000 per year for the next four years.

After accounting for these deductions, we will figure out how long the remaining amount can support your graduate school costs of $25,620 per year, considering that the funds are earning 8% interest annually.

### Step-by-Step Breakdown

1. **Remaining Fund After Giving $24,000 to Courtney**:
   \[
   \text{Remaining fund} = 150,000 - 24,000 = 126,000
   \]

2. **Yearly Vacation Deduction for Four Years**:
   Each year, $9,000 is deducted for vacations, but the remaining funds will also earn 8% interest annually. So, we need to calculate the future value of the fund after accounting for the vacations.

We will calculate the present value of the vacations over the four years, discounted at an 8% interest rate, and subtract it from the remaining fund. Let's compute that.

After giving $24,000 to your sister and accounting for the four yearly vacations (with a total present value of $29,809.14), the remaining fund is approximately **$96,190.86**.

Next, we'll calculate how long this remaining fund of $96,190.86 can cover your graduate school costs of $25,620 per year, considering an 8% annual interest rate. We'll use an annuity calculation for this.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 the steps manually or help with any other questions. Let me know how you'd like to proceed! 

In the meantime, here are some related questions that could help deepen your understanding of this topic:

1. How is the present value of future expenses calculated with interest rates?
2. What impact does compounding interest have on savings over multiple years?
3. How do annuities work when withdrawing from a fund at a fixed interest rate?
4. How would changing the interest rate affect the number of years you can stay in school?
5. If graduate school costs increased annually, how would that affect the duration of support?

**Tip:** When dealing with funds over time, consider both the costs and the interest gained, as it helps maximize the longevity of the funds.

Learn how to calculate the remaining amount from a $150,000 nest egg after deducting funds for a sibling's business and yearly vacations. This problem uses present value of annuities and compound interest formulas to find out how long the remaining funds can support yearly graduate school costs of $25,620. Suitable for Grades 11-12 or college students.

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