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. How much money will your parents have at the end of four years to help you with graduate school, which you will start then? 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 can break the problem into two main parts: 

1. **Calculating how much money will remain after 4 years** (considering the initial nest egg, the amount given to Courtney, and the vacation expenses).
2. **Determining how long that remaining amount will last for graduate school expenses**.

### Part 1: Calculating the Remaining Funds after 4 Years

- **Initial nest egg**: $150,000
- **Money given to Courtney**: $24,000 (subtracted from the initial amount)
- **Vacation expenses**: $9,000 per year for 4 years

The interest rate is 8% annually, which we will account for when calculating the future value of the money over 4 years.

#### Step 1: Remaining amount after giving Courtney her $24,000
\[
\text{Remaining amount} = 150,000 - 24,000 = 126,000
\]

#### Step 2: Adjusting for vacation expenses
We need to subtract $9,000 at the end of each year for 4 years. Each $9,000 payment will accrue different amounts of interest depending on when it was spent. We'll use the **future value** of each vacation expense to calculate its impact on the final sum. The formula for future value is:

\[
FV = PV \times (1 + r)^n
\]
Where:
- $PV$ is the payment,
- $r$ is the annual interest rate (8% or 0.08),
- $n$ is the number of years.

Let’s calculate the future value of the four vacation payments:
- The first vacation occurs at the end of year 1, so it will accumulate interest for 3 more years.
- The second vacation occurs at the end of year 2, so it will accumulate interest for 2 more years.
- The third vacation occurs at the end of year 3, so it will accumulate interest for 1 more year.
- The fourth vacation occurs at the end of year 4, so it will not accumulate any interest.

I'll now calculate the total future value of these expenses and how much will be left at the end of 4 years.It seems the calculation timed out. Let me redo the computations for the future value of the remaining funds manually.

### Step 2: Future Value of Vacation Expenses
For each vacation expense, we'll calculate the future value:

1. **Future Value of 1st Vacation (after Year 1)**:
   \[
   FV = 9,000 \times (1 + 0.08)^3 = 9,000 \times 1.259712 = 11,337.41
   \]
   
2. **Future Value of 2nd Vacation (after Year 2)**:
   \[
   FV = 9,000 \times (1 + 0.08)^2 = 9,000 \times 1.1664 = 10,497.60
   \]
   
3. **Future Value of 3rd Vacation (after Year 3)**:
   \[
   FV = 9,000 \times (1 + 0.08)^1 = 9,000 \times 1.08 = 9,720.00
   \]
   
4. **Future Value of 4th Vacation (after Year 4)**:
   \[
   FV = 9,000 \times (1 + 0.08)^0 = 9,000
   \]

### Total Future Value of Vacation Expenses:
\[
\text{Total Vacation Expenses} = 11,337.41 + 10,497.60 + 9,720.00 + 9,000 = 40,555.01
\]

### Step 3: Future Value of the Remaining Nest Egg
After giving Courtney $24,000, the remaining amount is $126,000. Now, let’s calculate the future value of this amount after 4 years at an 8% interest rate:

\[
FV = 126,000 \times (1 + 0.08)^4 = 126,000 \times 1.36048896 = 171,418.61
\]

### Step 4: Final Remaining Amount after Vacation Expenses
\[
\text{Final Amount} = 171,418.61 - 40,555.01 = 130,863.60
\]

### Part 2: How Long Can You Stay in Graduate School?

Graduate school costs $25,620 per year. So, we can divide the final remaining amount by the annual cost to determine how many years the funds will last:

\[
\text{Years in School} = \frac{130,863.60}{25,620} \approx 5.11 \text{ years}
\]

### Final Answer:
- After 4 years, your parents will have approximately **$130,863.60**.
- Based on these funds, you will be able to stay in school for about **5.11 years**.

Let me know if you want more details or clarifications.

---

Here are 5 questions to extend this scenario:
1. What if the interest rate were 6% instead of 8%?
2. How would the result change if vacation costs increased to $10,000 per year?
3. What if Courtney were given $30,000 instead of $24,000?
4. How would the timeline change if the cost of graduate school increased by 5% annually?
5. What happens if your graduate school costs were reduced by $5,000 per year?

**Tip:** Always consider future value when planning for long-term investments, as compound interest significantly impacts savings over time.

This math problem calculates how much of a $150,000 nest egg will remain after four years of 8% interest growth, factoring in $24,000 given to a sibling and $9,000 yearly vacations. The final amount is used to determine how long graduate school funds will last at a cost of $25,620 per year.

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