Suppose you wish to have $9,000 in 15 years. Use the present value formula to find how much you should invest now (in $) at 6% interest compounded semiannually in order to meet your goal. (Round your answer to the nearest cent.)
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The formula for the present value $ P $ of an investment, given the future value $ A $, interest rate $ r $, number of times the interest is compounded per year $ n $, and the time period in years $ t $, is:

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

Where:
- $ A = 9000 $ (the future value you want),
- $ r = 0.06 $ (6% annual interest rate),
- $ n = 2 $ (since the interest is compounded semiannually),
- $ t = 15 $ (the number of years).

Now, we can plug in the values and calculate the present value $ P $:

\[
P = \frac{9000}{(1 + \frac{0.06}{2})^{2 \times 15}}
\]

Let me calculate this for you.The amount you should invest now is approximately **$3,707.88** to meet your goal of $9,000 in 15 years at a 6% interest rate compounded semiannually.

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How does changing the compounding frequency affect the present value?
2. What happens if the interest rate is increased to 7%?
3. How would you calculate the future value if you invest $3,707.88 at 6% for 20 years instead of 15?
4. What would be the impact of monthly compounding instead of semiannually?
5. Can the present value formula be used for variable interest rates?

**Tip:** Increasing the number of compounding periods per year (e.g., monthly instead of semiannually) results in a slightly lower required investment due to more frequent interest accumulation.

Suppose you wish to have $9,000 in 15 years. Use the present value formula to find how much you should invest now (in $) at 6% interest compounded semiannually in order to meet your goal. (Round your answer to the nearest cent.)

Learn how to calculate the present value needed to reach $9,000 in 15 years with a 6% interest rate compounded semiannually. This problem covers compound interest concepts and the present value formula. Suitable for Grades 11-12 students.

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