How much money should be invested in an account that earns 8% interest compounded quarterly to yield $10,000 in 3 years? (Round your answer to the nearest cent.

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

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

Where:
- $A$ is the future value (the amount you want, which is $10,000),
- $P$ is the principal (the amount you need to invest initially),
- $r$ is the annual interest rate (8% or 0.08),
- $n$ is the number of times the interest is compounded per year (quarterly, so 4),
- $t$ is the time in years (3 years).

We need to solve for $P$. Rearranging the formula to solve for $P$:

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

Substituting the known values:

\[
P = \frac{10,000}{\left(1 + \frac{0.08}{4}\right)^{4 \times 3}}
\]

Now let’s calculate it.The initial investment required is approximately \$7,884.93 to yield \$10,000 in 3 years at an 8% interest rate compounded quarterly.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the frequency of compounding affect the amount to be invested?
2. What would happen if the interest rate were 6% instead of 8%?
3. How much interest is earned over the 3 years in this example?
4. What would be the required investment for 5 years instead of 3 years?
5. How would the initial investment change if the interest was compounded annually?

**Tip:** The more frequently interest is compounded, the more the investment grows, as interest is calculated on accumulated interest as well.

How much money should be invested in an account that earns 8% interest compounded quarterly to yield $10,000 in 3 years? (Round your answer to the nearest cent.)

Learn how to calculate the initial investment required to yield $10,000 in 3 years with 8% interest compounded quarterly. This problem demonstrates the use of the compound interest formula and exponential growth in finance mathematics. Suitable for students in Grades 10-12.

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