You need to have $50,000 saved in 20 years. If your account earns 5.6% compounded semiannually, how much would you need to deposit now to reach your goal? Round your answer to the nearest cent.

To determine how much you need to deposit now, we can use the formula for compound interest, specifically for a future value (FV):

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

Where:
- $FV$ is the future value, which is $50,000.
- $P$ is the present value or the amount you need to deposit now (this is what we're solving for).
- $r$ is the annual interest rate, which is 5.6%, or 0.056 as a decimal.
- $n$ is the number of times the interest is compounded per year. Since it's compounded semiannually, $n = 2$.
- $t$ is the number of years, which is 20 years.

Rearranging the formula to solve for $P$ (the initial deposit):

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

Now, we can substitute the values:

\[
P = \frac{50,000}{\left(1 + \frac{0.056}{2}\right)^{2 \times 20}}
\]

Let me calculate this for you.The amount you would need to deposit now to reach your goal of $50,000 in 20 years, with an interest rate of 5.6% compounded semiannually, is approximately **$16,567.06**.

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

Here are 5 related questions to consider:
1. How would the required deposit change if the interest rate were 6% instead of 5.6%?
2. What if the interest was compounded quarterly instead of semiannually?
3. How does changing the time period to 15 years affect the initial deposit?
4. What is the difference between simple interest and compound interest in this context?
5. How much interest will you earn over the 20 years?

**Tip:** The more frequently interest is compounded, the greater the amount of interest accrued over time!

Learn how to calculate the initial deposit required to save $50,000 in 20 years with an interest rate of 5.6% compounded semiannually. This financial math problem demonstrates the use of the compound interest formula to solve for present value. Suitable for Grades 10-12.

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