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The problem asks to find the present value $ P $ that will amount to $260,000 if it is invested at 6% annual interest, compounded semiannually, for 18 years. To solve this, we can use the **Present Value (PV)** formula for compound interest:

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

Where:
- $ A = 260,000 $ is the future value (the amount we want to reach),
- $ r = 0.06 $ (6%) is the annual interest rate,
- $ n = 2 $ is the number of compounding periods per year (since it's compounded semiannually),
- $ t = 18 $ is the time in years.

Now, let's plug the values into the formula:

\[
P = \frac{260,000}{\left(1 + \frac{0.06}{2}\right)^{2 \times 18}}
\]
\[
P = \frac{260,000}{\left(1 + 0.03\right)^{36}} = \frac{260,000}{1.03^{36}}
\]

Next, calculate $ 1.03^{36} $.

\[
1.03^{36} \approx 2.8983
\]

Now, divide the future value by this factor:

\[
P = \frac{260,000}{2.8983} \approx 89,707.58
\]

Thus, the present value $ P $ is approximately **$89,707.58**.

Would you like more details or have any questions about this?

### Related Questions:
1. What would the present value be if the interest rate were 5% instead of 6%?
2. How would the answer change if the investment period was 20 years instead of 18 years?
3. What is the future value if $100,000 is invested at 6% compounded semiannually for 18 years?
4. How does the number of compounding periods affect the present value?
5. What is the formula for continuous compounding, and how does it compare to semiannual compounding?

### Tip:
The higher the frequency of compounding (e.g., monthly instead of semiannually), the more interest is earned over time, resulting in a smaller present value for the same future goal.

What present value P amounts to $260,000 if it is invested at 6%, compounded semiannually, for 18 years?

Learn how to calculate the present value needed to reach $260,000 in 18 years with a 6% annual interest rate compounded semiannually. The solution uses the compound interest formula, making it a valuable resource for high school students.

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