How much money should be deposited today in an account that earns 8 %8% compounded semiannually so that it will accumulate to $ 9000$9000 in three​ years?
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Part 1
The amount of money that should be deposited is ​$ enter your response here.
​(Round up to the nearest​ cent.)

To find how much money should be deposited today, we can use the formula for compound interest:

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

Where:
- $ A $ is the future amount (the accumulated amount, which is $9000),
- $ P $ is the present value (the amount to be deposited today),
- $ r $ is the annual interest rate (8% = 0.08),
- $ n $ is the number of times the interest is compounded per year (semiannually means $ n = 2 $),
- $ t $ is the number of years (3 years).

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

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

Now, let's plug in the values:

\[
P = \frac{9000}{\left(1 + \frac{0.08}{2}\right)^{2 \times 3}}
\]

Let's calculate it.

The amount of money that should be deposited today is approximately **$7,112.83**.

Would you like further clarification or details on any step?

Here are 5 related questions to explore:
1. How would the result change if the interest were compounded quarterly instead of semiannually?
2. What would be the future value if $7,112.83 were invested for 5 years instead of 3 at the same interest rate?
3. How does changing the interest rate to 10% affect the deposit required for $9,000 in 3 years?
4. What is the total interest earned on the $7,112.83 deposit over 3 years?
5. How would the formula change if interest were compounded continuously?

**Tip:** In finance, the number of compounding periods significantly affects how fast your investment grows.

How much money should be deposited today in an account that earns 8% compounded semiannually so that it will accumulate to $9000 in three years?

Find out how much you need to deposit today to accumulate $9000 in 3 years with an 8% interest rate compounded semiannually. This compound interest problem involves calculating the present value using key finance formulas. Perfect for high school students in Grades 10-12.

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