How much money should be invested in an account that earns 9% interest compounded quarterly to yield $10,000 in 7 years? (Round your answer to the nearest cent.)
$  

To determine how much money should be invested, we can use the formula for compound interest:

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

Where:
- $ A $ is the future value (the amount of money you want, which is $10,000),
- $ P $ is the initial principal (the amount to be invested, which we are trying to find),
- $ r $ is the annual interest rate (9%, or 0.09 as a decimal),
- $ n $ is the number of times the interest is compounded per year (quarterly means $ n = 4 $),
- $ t $ is the time in years (7 years).

We need to solve for $ P $:

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

Let's plug in the values and calculate it:

\[
P = \frac{10000}{\left(1 + \frac{0.09}{4}\right)^{4 \times 7}}
\]

Now, I'll calculate it for you.The amount that should be invested is **$5,363.24** to yield $10,000 in 7 years with a 9% interest rate compounded quarterly.

Would you like further explanation or have any questions?

Here are 5 related questions:
1. How does the frequency of compounding affect the future value of an investment?
2. What would the initial investment be if the interest rate were 6% instead of 9%?
3. How much interest would be earned over the 7 years on the $5,363.24 investment?
4. What is the formula for continuous compounding, and how would the answer change using it?
5. How would the required investment change if the target amount was $20,000 instead of $10,000?

**Tip:** Compounding more frequently (e.g., quarterly vs. annually) will increase the amount of interest earned over time!

How much money should be invested in an account that earns 9% interest compounded quarterly to yield $10,000 in 7 years? (Round your answer to the nearest cent.)

Find out how much to invest in an account earning 9% interest compounded quarterly to yield $10,000 in 7 years. This detailed solution uses the compound interest formula, making it ideal for high school students learning about exponential growth and finance.

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