Your younger sister, Linda, will start college in five years. She has just informed your parents that she wants to go to Hampton University, which will cost $41,000 per year for four years (cost assumed to come at the end of each year). Anticipating Linda’s ambitions, your parents started investing $6,100 per year five years ago and will continue to do so for five more years. Use 11 percent as the appropriate interest rate throughout this problem (for discounting or compounding).

How much will your parents have to save each year (A?) for the next five years in addition to the $6,100 they are currently saving to have the necessary funds for Linda's education? Use Appendix C and Appendix D for an approximate answer, but calculate your final answer using the formula and financial calculator methods.

To solve this problem, we'll break it down step by step.

### Step 1: Determine the future cost of Linda’s college education.

The college cost is $41,000 per year for 4 years, and the payments are assumed to be made at the end of each year. Since we’re dealing with a future series of payments, we will calculate the present value (PV) of these payments at the start of Linda’s college (five years from now) and then work backward to determine how much the parents need to save.

The formula for the present value of an annuity (for calculating the present value of college costs) is:

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

Where:
- $ C = 41,000 $ (annual college cost),
- $ r = 0.11 $ (interest rate of 11%),
- $ n = 4 $ (number of years of college payments).

Let’s calculate this first.

### Step 2: Calculate the future value of the current savings.

Your parents have been saving $6,100 per year for the last 5 years and will continue to do so for the next 5 years. This represents two separate series of cash flows:
1. The future value of the first 5 payments already made.
2. The future value of the next 5 payments they will make.

The formula for the future value of an annuity is:

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

Where:
- $ P = 6,100 $ (annual savings),
- $ r = 0.11 $ (interest rate),
- $ n = 5 $ (number of years).

We’ll calculate the future value of both the past and future savings.

### Step 3: Determine the additional savings required.

Finally, we will subtract the future value of the savings (from Step 2) from the present value of the college cost (from Step 1) to find the total amount needed. Then, we will calculate how much additional savings (denoted as $ A $) per year for the next 5 years is required.

Let me now calculate these steps for you.Here are the calculations based on the given information:

1. **Present Value of College Costs**: The present value of the total cost for Linda’s college education, starting five years from now, is approximately **$127,200.27**.

2. **Future Value of Past Savings**: The future value of the $6,100 saved each year for the past five years, compounded over the next five years, is approximately **$42,168.44**.

3. **Future Value of Future Savings**: The future value of the $6,100 saved each year for the next five years is approximately **$37,989.59**.

4. **Additional Savings Needed**: The total additional amount required after accounting for current savings is approximately **$47,042.24**.

5. **Additional Annual Savings (A)**: To meet the remaining required amount, your parents will need to save an additional **$7,553.59** per year for the next five years.

### Summary:
In addition to their current $6,100 annual savings, your parents need to save an extra $7,553.59 per year for the next five years to cover the cost of Linda's education.

Do you have any questions or would you like more details on any step?

Here are some related questions:
1. How does compounding affect savings over time?
2. What is the impact of increasing the interest rate on future savings?
3. How would lowering the college cost affect the required additional savings?
4. What happens if your parents extend the savings period beyond five years?
5. How do other investment options compare at different interest rates?

**Tip**: Regularly reviewing your savings plan can help adjust for changing expenses and financial goals.

Your younger sister, Linda, will start college in five years. She has just informed your parents that she wants to go to Hampton University, which will cost $41,000 per year for four years (cost assumed to come at the end of each year). Anticipating Linda’s ambitions, your parents started investing $6,100 per year five years ago and will continue to do so for five more years. Use 11 percent as the appropriate interest rate throughout this problem (for discounting or compounding). How much will your parents have to save each year (A?) for the next five years in addition to the $6,100 they are currently saving to have the necessary funds for Linda's education? Use Appendix C and Appendix D for an approximate answer, but calculate your final answer using the formula and financial calculator methods.

Learn how to calculate the additional savings required for Linda's college education. This problem involves calculating the present and future value of annuities, using an 11% interest rate over five years. Includes step-by-step formulas and financial calculations for planning college expenses.

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