You wish to have $23,000 in 8 years. Use Table 11-2 to create a new table factor, and then find how much you should invest now (in $) at 6% interest, compounded quarterly in order to have $23,000, 8 years from now. (Round your answer to the nearest cent.)
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To determine how much you should invest now to reach $23,000 in 8 years, with 6% interest compounded quarterly, we can use the formula for **Present Value (PV)** of a future sum, which is:

\[
PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}
\]

Where:
- $ PV $ is the present value (the amount you need to invest now),
- $ FV $ is the future value (\$23,000),
- $ r $ is the annual interest rate (6% or 0.06),
- $ n $ is the number of times the interest is compounded per year (quarterly, so $ n = 4 $),
- $ t $ is the time in years (8 years).

### Step-by-step:

1. **Identify the known values:**
   - $ FV = 23,000 $
   - $ r = 0.06 $
   - $ n = 4 $
   - $ t = 8 $

2. **Plug the values into the formula:**
   \[
   PV = \frac{23,000}{(1 + \frac{0.06}{4})^{4 \times 8}}
   \]

3. **Calculate the present value.**

Let me calculate this for you.The amount you should invest now to have \$23,000 in 8 years, at a 6% interest rate compounded quarterly, is approximately **\$14,282.84**.

Would you like more details or have any questions on this process?

Here are 5 related questions for further exploration:
1. How would the required investment change if the interest rate was 8% instead of 6%?
2. What happens to the required investment if the interest is compounded monthly instead of quarterly?
3. How much would you have if you invested \$14,282.84 at the same rate for 10 years instead of 8 years?
4. What is the difference in total interest earned between quarterly and yearly compounding over 8 years?
5. How does increasing the compounding frequency affect the total amount after 8 years?

**Tip:** Compounding more frequently results in slightly higher future values, as interest is applied more often, though the effect diminishes as the compounding frequency increases.

You wish to have $23,000 in 8 years. Use Table 11-2 to create a new table factor, and then find how much you should invest now (in $) at 6% interest, compounded quarterly in order to have $23,000, 8 years from now. (Round your answer to the nearest cent.)

Learn how to calculate the present value of an investment to reach $23,000 in 8 years with 6% interest compounded quarterly. This detailed explanation covers key concepts in finance, including the Present Value and Future Value formulas, suitable for high school students in Grades 10-12.

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