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The problem you're working on involves calculating the future value of an annuity for scholarships. Let's break it down step by step:

### Problem Description
The company invests **$10,000 every 3 months (quarterly)** in an annuity that pays **10.5% interest compounded quarterly**. We need to find:

1. **(a)** How much the company will have in scholarship funds at the end of **10 years**.
2. **(b)** The total interest earned.

### Approach
This is a **future value of an ordinary annuity** problem, where periodic investments are made, and the interest is compounded. The formula for the future value (FV) of an ordinary annuity is:

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

Where:
- $ P $ is the periodic investment (in this case, $10,000).
- $ i $ is the interest rate per period (quarterly interest rate).
- $ n $ is the total number of periods (number of quarters over 10 years).

#### Step 1: Define Variables
- $ P = 10,000 $ (investment per quarter)
- Annual interest rate is 10.5%, so the quarterly interest rate $ i = \frac{10.5\%}{4} = 0.02625 $
- 10 years, compounded quarterly, means $ n = 10 \times 4 = 40 $ quarters.

#### Step 2: Calculate Future Value
Now, let's plug these values into the formula to find the future value. Let me calculate this.The future value (FV) of the scholarship fund at the end of 10 years is **$693,030.84**.

### Step 3: Find the Interest Earned
The interest earned is the total amount accumulated minus the total amount invested. The total amount invested over 10 years is:

\[
\text{Total investment} = P \times n = 10,000 \times 40 = 400,000
\]

The interest is then:

\[
\text{Interest} = FV - \text{Total investment}
\]

Let me calculate that for you.The total interest earned over 10 years is **$293,030.84**.

### Summary:
1. **Future value of the scholarship fund after 10 years**: **$693,030.84**
2. **Total interest earned**: **$293,030.84**

Would you like further details or explanations on any part of this process?

Here are some related questions to consider:
1. What would happen if the interest rate increased to 12%?
2. How much would the future value be if the investment period extended to 15 years?
3. How does switching to monthly compounding affect the future value?
4. What is the present value of the scholarship fund if the company wanted the same future value today?
5. How does the future value change if the company invests $15,000 every quarter instead?

**Tip**: Always double-check if the interest rate is compounded periodically (quarterly, monthly, etc.) to apply the correct formula for the future value.

To offer scholarships to children of employees, a company invests $10,000 at the end of every three months in an annuity that pays 10.5% compounded quarterly. How much will the company have in scholarship funds at the end of 10 years? Find the interest.

Learn how to calculate the future value of a quarterly investment of $10,000 over 10 years with a 10.5% interest rate compounded quarterly. The future value and total interest earned are computed using the future value of an annuity formula.

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