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. Do not round until the final answer, then round up to the nearest dollar as needed.

To solve this problem, we need to use the **future value of an ordinary annuity** formula. This formula is used to determine the amount of each periodic deposit when the future value (the financial goal) is known. The formula is:

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

Where:
- $ FV $ = future value or financial goal = $15,000
- $ P $ = periodic deposit (which we need to determine)
- $ r $ = interest rate per period
- $ n $ = total number of periods

### Step-by-step solution:

#### 1. **Determine the values for each variable:**
   - **Interest rate per period**: The rate is 2.5% compounded quarterly, so the quarterly rate $ r $ is:
     \[
     r = \frac{2.5\%}{4} = \frac{0.025}{4} = 0.00625
     \]
   - **Number of periods**: Since the deposits are made every three months (quarterly), and the time is 4 years, the total number of periods $ n $ is:
     \[
     n = 4 \text{ years} \times 4 \text{ quarters per year} = 16 \text{ periods}
     \]
   
#### 2. **Substitute these values into the future value formula:**

\[
15,000 = P \times \frac{(1 + 0.00625)^{16} - 1}{0.00625}
\]

#### 3. **Solve for the periodic deposit $ P $:**

First, calculate $ (1 + 0.00625)^{16} $:

\[
(1.00625)^{16} \approx 1.104941
\]

Now calculate $ \frac{1.104941 - 1}{0.00625} $:

\[
\frac{1.104941 - 1}{0.00625} = \frac{0.104941}{0.00625} \approx 16.79056
\]

Now, substitute this into the equation:

\[
15,000 = P \times 16.79056
\]

Solve for $ P $:

\[
P = \frac{15,000}{16.79056} \approx 893.31
\]

So, the periodic deposit $ P $ is approximately **$894** when rounded to the nearest dollar.

### Part b. How much of the financial goal comes from deposits and how much comes from interest?

- **Total deposits**: 
The total number of deposits made is 16, and the deposit amount is approximately $894:
\[
\text{Total deposits} = 16 \times 894 = 14,304
\]

- **Interest earned**: 
The interest is the difference between the financial goal and the total deposits:
\[
\text{Interest} = 15,000 - 14,304 = 696
\]

Thus, $696 of the financial goal comes from interest, and $14,304 comes from the deposits.

---

Would you like more details or have any questions? Here are 5 related questions for further practice:

1. How would the periodic deposit change if the interest rate were higher?
2. What is the effect of compounding more frequently, such as monthly instead of quarterly?
3. How much would the final amount be if the period were 5 years instead of 4 years?
4. What would be the interest portion if the deposits were made monthly instead of quarterly?
5. How would the financial goal change if you deposit $800 quarterly for 4 years?

**Tip**: When calculating compound interest, the more frequently the interest is compounded, the more the future value grows!

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: $15,000. Determine the periodic deposit in dollars. Do not round until the final answer, then round up to the nearest dollar as needed.

Calculate the periodic deposit required to reach a $15,000 financial goal over 4 years with a 2.5% interest rate compounded quarterly. Learn how to apply the future value of an ordinary annuity formula and find out how much of the final amount comes from deposits versus interest.

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