a. Use the appropriate formula to determine the periodic deposit. b. How
 much of the financial goal comes from deposits and how much comes from 
interest? Data is as follows. Periodic deposit: $? at the end of every 
three months, rate: 2.5% compounded quarterly, time: 4 years, financial 
goal: $15000. Determine the periodic deposit in dollars.

We can solve this problem using the **Future Value of an Annuity** formula, since we are dealing with regular, periodic deposits made at the end of each period. The formula for the future value of an annuity (with compound interest) is:

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

Where:
- $FV$ is the future value or the financial goal (in this case, $15,000),
- $P$ is the periodic deposit (the unknown value we are solving for),
- $r$ is the interest rate per period (quarterly in this case),
- $n$ is the total number of periods (quarters).

### Step 1: Calculate the necessary components.

- **Interest rate per period ($r$):**
  Since the annual interest rate is 2.5% and the interest is compounded quarterly, the interest rate per quarter is:
  \[
  r = \frac{2.5\%}{4} = \frac{0.025}{4} = 0.00625
  \]

- **Total number of periods ($n$):**
  There are 4 quarters in a year, and the time period is 4 years, so:
  \[
  n = 4 \times 4 = 16 \text{ quarters}
  \]

### Step 2: Solve for $P$ (the periodic deposit).

We rearrange the future value formula to solve for $P$:
\[
P = \frac{FV \times r}{(1 + r)^n - 1}
\]

Substituting the known values:
\[
P = \frac{15000 \times 0.00625}{(1 + 0.00625)^{16} - 1}
\]

Let's calculate this value.

The periodic deposit required to reach the financial goal of $15,000 is approximately **$894.33**.

### Part b: Breakdown of the financial goal
We can now determine how much of the $15,000 comes from the total deposits and how much comes from interest. The total amount deposited is simply the number of deposits times the periodic deposit, and the rest is from interest.

- **Total amount from deposits:**
  \[
  \text{Total Deposits} = P \times n = 894.33 \times 16
  \]
- **Amount from interest:**
  \[
  \text{Interest} = FV - \text{Total Deposits}
  \]

Let's calculate these values.- **Total amount from deposits**: $14,309.29
- **Amount from interest**: $690.71

So, out of the $15,000 financial goal:
- $14,309.29 comes from your deposits,
- $690.71 comes from the interest earned.

Would you like further details or have any questions?

### Follow-up Questions:
1. What happens if the interest rate changes to 3%?
2. How does increasing the deposit period to 5 years impact the final amount?
3. How much would you need to deposit monthly instead of quarterly?
4. What if the deposits are made at the beginning of each period instead of the end?
5. What would the interest be if the rate was compounded monthly instead?

### Tip:
When planning for financial goals, try to factor in inflation, as it may affect the real value of the future amount.

Use the appropriate formula to determine the periodic deposit. How much of the financial goal comes from deposits and how much comes from interest? Data is as follows: Periodic deposit: $? at the end of every three months, rate: 2.5% compounded quarterly, time: 4 years, financial goal: $15000. Determine the periodic deposit in dollars.

Learn how to calculate the periodic deposit needed to reach a financial goal of $15,000 with 2.5% interest compounded quarterly over 4 years. This solution uses the Future Value of an Annuity formula and provides a breakdown of how much comes from deposits versus interest. Suitable for students in Grades 10-12.

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