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?

Periodic Deposit:$? at the end of every three months

Rate:4.5% compounded quarterly

Time: 4 years

Financial Goal:$25,000

To solve this problem, we need to determine the periodic deposit for an ordinary annuity where interest is compounded quarterly. The formula for calculating the periodic deposit ($ R $) in such a scenario is:

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

Where:
- $ A $ is the financial goal ($25,000),
- $ i $ is the interest rate per period,
- $ n $ is the total number of periods.

### Step 1: Identify the variables
- **Financial Goal (A):** $25,000
- **Annual Interest Rate:** 4.5% (0.045 as a decimal)
- **Compounding Periods per Year:** 4 (quarterly)
- **Time (T):** 4 years

### Step 2: Calculate the periodic interest rate $ i $
\[
i = \frac{0.045}{4} = 0.01125
\]

### Step 3: Calculate the total number of periods $ n $
\[
n = 4 \, (\text{periods per year}) \times 4 \, (\text{years}) = 16
\]

### Step 4: Apply the formula to find the periodic deposit ($ R $)

\[
R = \frac{25,000 \times 0.01125}{(1 + 0.01125)^{16} - 1}
\]

Now, let's compute this.The periodic deposit needed is approximately **$1,434.84** at the end of every three months.

### Part b: How much comes from deposits and how much from interest?

- **Total deposits:** To find the total amount deposited, we multiply the periodic deposit by the number of periods:

\[
\text{Total Deposits} = R \times n = 1,434.84 \times 16
\]

- **Interest earned:** Subtract the total deposits from the financial goal to find the amount that comes from interest.

Let me calculate these values for you.- **Total deposits:** $22,957.45
- **Interest earned:** $2,042.55

Thus, out of the $25,000 financial goal:
- $22,957.45 comes from your deposits.
- $2,042.55 comes from the interest earned over the 4 years.

Would you like more details or have any questions?

### Related Questions:
1. How does changing the interest rate affect the periodic deposit?
2. What happens to the total interest earned if the time increases to 5 years?
3. How would the periodic deposit change if the compounding frequency is monthly instead of quarterly?
4. What if the financial goal were $30,000 instead of $25,000—how would the results differ?
5. How is the formula adjusted for continuous compounding?

### Tip:
The more frequent the compounding periods, the more interest you earn on interest already accrued, which is known as **compound interest effect**.

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? Periodic Deposit:$? at the end of every three months Rate:4.5% compounded quarterly Time: 4 years Financial Goal:$25,000

Learn how to calculate the periodic deposit needed to reach a financial goal of $25,000 over 4 years with a 4.5% interest rate compounded quarterly. This problem covers key concepts in annuities and compound interest, providing a step-by-step solution to determine deposits and interest earned.

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