How much money should be invested in an account that earns 8% 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 (the present value $ P $) in an account that earns 8% interest compounded quarterly to yield $10,000 in 7 years, we can use the compound interest formula:

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

Where:
- $ A $ is the final amount ($10,000),
- $ P $ is the initial investment (the amount to find),
- $ r $ is the annual interest rate (8% or 0.08),
- $ n $ is the number of times the interest is compounded per year (quarterly, so $ n = 4 $),
- $ t $ is the time in years (7 years).

Now, solving for $ P $:

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

Substitute the values:

\[
P = \frac{10,000}{\left(1 + \frac{0.08}{4}\right)^{4 \times 7}}
\]

Let me calculate this.The amount that should be invested to yield $10,000 in 7 years, at an 8% interest rate compounded quarterly, is approximately **$5,743.75**.

Would you like further details on this, or have any other questions?

Here are 5 related questions to explore further:
1. How does changing the compounding frequency affect the investment required?
2. What would the investment be if the interest rate were compounded monthly instead of quarterly?
3. How much interest would be earned over the 7 years?
4. How does the formula change if the interest is compounded continuously?
5. How much would you need to invest to get $20,000 instead?

**Tip**: Compounding more frequently results in a lower initial investment for the same final amount due to the more frequent application of interest.

How much money should be invested in an account that earns 8% interest compounded quarterly to yield $10,000 in 7 years? (Round your answer to the nearest cent.)

Find out how much you need to invest today at an 8% interest rate compounded quarterly to reach $10,000 in 7 years. Learn the formula for compound interest and present value, suitable for students in Grades 10-12.

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