The management of Gibraltar Brokerage Services anticipates a capital expenditure of $23,000 in 4 years for the purchase of new computers and has decided to set up a sinking fund to finance this purchase. If the fund earns interest at the rate of 2%/year compounded quarterly, determine the size of each (equal) quarterly installment that should be deposited in the fund. (Round your answer to the nearest cent.)

We are dealing with a sinking fund problem, where we need to determine the amount of each equal quarterly installment needed to accumulate a future value of $23,000 over 4 years, with a 2% annual interest rate compounded quarterly.

### Step-by-step solution:

1. **Future Value Formula for Sinking Fund:**

   The formula for the future value of a sinking fund is:

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

   Where:
   - $A$ is the future value (amount needed, $23,000),
   - $P$ is the amount of each payment (what we are solving for),
   - $r$ is the interest rate per period (quarterly interest rate),
   - $n$ is the total number of periods (number of quarters).

2. **Identify the values:**
   - Future Value $A = 23,000$,
   - Annual interest rate = 2%, so quarterly interest rate $r = \frac{2\%}{4} = 0.005$,
   - Number of periods $n = 4 \text{ years} \times 4 = 16 \text{ quarters}$.

3. **Rearranging the formula:**

   Solve for $P$ (the quarterly payment):

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

   Substituting the known values:

   \[
   P = \frac{23,000}{\frac{(1 + 0.005)^{16} - 1}{0.005}}
   \]

4. **Calculate the values:**

Now, I'll calculate the value of $P$.The size of each quarterly installment should be approximately **$1,384.36** to accumulate $23,000 in 4 years, with an interest rate of 2% compounded quarterly.

Would you like more details or have any other questions? Here are some related questions to explore:

1. How would the installment change if the interest rate was 3% instead of 2%?
2. What is the total amount paid into the fund over the 4 years?
3. How much interest is earned over the 4-year period?
4. How would the installment change if the goal was reached in 5 years instead of 4?
5. How does quarterly compounding affect the overall interest compared to annual compounding?

**Tip:** The higher the frequency of compounding, the more interest you earn over time.

The management of Gibraltar Brokerage Services anticipates a capital expenditure of $23,000 in 4 years for the purchase of new computers and has decided to set up a sinking fund to finance this purchase. If the fund earns interest at the rate of 2%/year compounded quarterly, determine the size of each (equal) quarterly installment that should be deposited in the fund.

Learn how to calculate the quarterly installment needed to accumulate $23,000 over 4 years with a 2% annual interest rate compounded quarterly. This problem involves sinking funds, compound interest, and the future value of annuities. Perfect for students in Grades 11-12 studying financial mathematics.

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