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To calculate the future value of an ordinary annuity with the given details, we’ll use the formula for the future value of an ordinary annuity (FV):

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

where:
- $ P $ is the payment amount,
- $ r $ is the interest rate per compounding period,
- $ n $ is the total number of compounding periods.

### Step-by-Step Solution

1. **Identify Variables:**
   - Payment ($ P $) = $4,500
   - Time period (in years) = 5
   - Nominal interest rate = 6%
   - Payment frequency = every 6 months, which means two payments per year.
   - Compounded semiannually (every 6 months).

2. **Determine the Period Rate and Number of Periods:**
   - Since the interest is compounded semiannually, the compounding period matches the payment frequency.
   - Convert the nominal rate to a per-period rate:  
     \[
     r = \frac{6\%}{2} = 3\% = 0.03
     \]
   - Total number of periods:
     \[
     n = 5 \text{ years} \times 2 \text{ periods per year} = 10 \text{ periods}
     \]

3. **Apply the Formula:**
   Substitute the values into the formula:
   \[
   FV = 4500 \times \left( \frac{(1 + 0.03)^{10} - 1}{0.03} \right)
   \]

4. **Calculate the Future Value:**
   - First, calculate $ (1 + 0.03)^{10} $:
     \[
     (1 + 0.03)^{10} \approx 1.34392
     \]
   - Now, subtract 1:
     \[
     1.34392 - 1 = 0.34392
     \]
   - Divide by 0.03:
     \[
     \frac{0.34392}{0.03} \approx 11.464
     \]
   - Finally, multiply by the payment amount ($4,500):
     \[
     FV \approx 4500 \times 11.464 = 51588
     \]

Therefore, the **future value of the annuity is approximately $51,588.00**.

Would you like further details or have any questions?

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### Further Exploration Questions
1. How would the future value change if the nominal interest rate was increased to 8%?
2. What if the annuity payments were made quarterly instead of semiannually?
3. How does an ordinary annuity differ from an annuity due in terms of future value calculation?
4. What would the future value be if the payments continued for 10 years instead of 5 years?
5. How is the future value of an annuity affected by compounding frequency?

### Tip
The more frequently interest is compounded, the greater the future value, due to the effects of compound interest over time.

Use Table 12-1 to calculate the future value (in $) of the ordinary annuity. Given: Annuity payment = $4,500 every 6 months, Time period = 5 years, Nominal rate = 6%, Compounded semiannually.

Learn how to calculate the future value of an ordinary annuity with semiannual payments of $4,500 over 5 years at a 6% nominal rate. This detailed solution walks through the step-by-step process, applying the future value formula and explaining key financial concepts such as compound interest.

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