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The problem you're working on involves finding the future value of an ordinary annuity, where the periodic payments are given along with the interest rate and compounding frequency. Here's the information extracted from the image:

- Payment amount per period, $ R = 13,000 $
- Interest rate, $ i = 4.2\% $ per year, compounded quarterly
- Time period, $ t = 9 $ years

The formula for the future value of an ordinary annuity is:

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

Where:
- $ R $ is the regular payment amount
- $ i $ is the annual interest rate (decimal form)
- $ n $ is the number of compounding periods per year (quarterly = 4)
- $ t $ is the total number of years

### Step 1: Calculate the quarterly interest rate
Since the interest is compounded quarterly:
\[
\frac{i}{n} = \frac{4.2\%}{4} = \frac{0.042}{4} = 0.0105
\]

### Step 2: Total number of periods
The number of periods for quarterly compounding over 9 years is:
\[
nt = 9 \times 4 = 36
\]

### Step 3: Plug values into the formula
Now we can compute the future value (FV):
\[
FV = 13,000 \times \frac{(1 + 0.0105)^{36} - 1}{0.0105}
\]

Let me calculate the value for you.The future value (FV) of the ordinary annuity is approximately **$565,177.83**.

Now, you also need to determine how much of this value is from contributions and how much is from interest:

### Contributions:
The total contributions over 9 years are:
\[
\text{Total Contributions} = R \times nt = 13,000 \times 36 = 468,000
\]

### Interest Earned:
The interest earned is the difference between the future value and the total contributions:
\[
\text{Interest Earned} = FV - \text{Total Contributions} = 565,177.83 - 468,000 = 97,177.83
\]

So, the breakdown is:
- Contributions: **$468,000**
- Interest earned: **$97,177.83**

Would you like more details or have any further questions?

Here are 5 related questions you might find useful:
1. What is the difference between ordinary annuity and annuity due?
2. How does changing the compounding frequency affect the future value?
3. What happens to the future value if the interest rate increases?
4. How can you calculate the present value of an annuity?
5. What are some practical uses of annuities in financial planning?

**Tip:** Always ensure that your interest rate and compounding frequency are aligned when applying annuity formulas.

Find the future value of an ordinary annuity if payments are made in the amount R and interest is compounded as given. Then determine how much of this value is from contributions and how much is from interest. R = 13,000; 4.2% interest compounded quarterly for 9 years.

Calculate the future value of an ordinary annuity with periodic payments of $13,000, compounded quarterly at 4.2% interest over 9 years. Learn how to break down the total amount into contributions and interest earned.

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