
Part 1
To offer scholarships to children of​ employees, a company invests ​$9000 at the end of every three months in an annuity that pays 11​% compounded quarterly.
a. How much will the company have in scholarship funds at the end of ten​ years?
b. Find the interest.

To solve this problem, we'll use the formulas for **future value** of an annuity and **interest calculation**.

### Given:
- **Periodic investment (PMT)** = $9000
- **Interest rate (r)** = 11% compounded quarterly, or 0.11/4 = 0.0275 per quarter
- **Number of periods per year** = 4 (quarterly compounding)
- **Time (t)** = 10 years
- **Total number of periods (n)** = $ 4 \times 10 = 40 $ quarters

### a. Future Value of the Annuity
The future value $ FV $ of an ordinary annuity (where payments are made at the end of each period) is given by the formula:

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

Where:
- $ PMT $ is the regular payment amount
- $ r $ is the interest rate per period
- $ n $ is the total number of periods

Substituting the values:

\[
FV = 9000 \times \left( \frac{(1 + 0.0275)^{40} - 1}{0.0275} \right)
\]

Now, let's calculate the future value.

### b. Interest Earned
The interest earned is the difference between the future value and the total amount invested.

\[
\text{Interest} = FV - (PMT \times n)
\]

Let me compute these values.### a. The future value of the scholarship fund at the end of 10 years is approximately **$641,413.30**.

### b. The total interest earned is approximately **$281,413.30**.

Would you like further details or clarifications on these calculations?

Here are some related questions to explore:
1. What happens if the interest rate changes to 12% compounded quarterly?
2. How does the future value change if the company invests for 15 years instead of 10?
3. What would be the total interest earned if the company invests $7000 quarterly instead of $9000?
4. How would the future value be affected if the payments were made at the beginning of each period instead?
5. How can the company calculate the present value required to reach a future value of $900,000 in 10 years?

**Tip:** The more frequent the compounding periods, the more interest you can accumulate over time, even with the same annual rate.

To offer scholarships to children of​ employees, a company invests ​$9000 at the end of every three months in an annuity that pays 11​% compounded quarterly.
 a. How much will the company have in scholarship funds at the end of ten​ years?
 b. Find the interest.

Learn how to calculate the future value and interest earned from a $9000 quarterly investment at 11% compounded quarterly over 10 years. This problem involves financial mathematics concepts like annuities and compound interest, using formulas for future value and interest calculation.

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