An investment company pays 3​% compounded semiannually. You want to have $ 16 comma 000 in the future.
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Part 1
​(A) How much should you deposit now to have that amount 5 years from​ now?


To solve this problem, we will use the compound interest formula:

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

Where:
- $A$ is the amount of money in the future (future value),
- $P$ is the principal amount (initial deposit),
- $r$ is the annual interest rate (expressed as a decimal),
- $n$ is the number of times the interest is compounded per year,
- $t$ is the number of years.

In this case:
- $A = 16,000$ (the future value you want),
- $r = 0.03$ (3% annual interest rate),
- $n = 2$ (since the interest is compounded semiannually),
- $t = 5$ years.

We need to find $P$, the amount you need to deposit now.

First, solve for $P$ by rearranging the formula:

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

Now, let's calculate $P$.The amount you should deposit now to have $16,000 in 5 years, with a 3% interest rate compounded semiannually, is approximately **$13,786.68**.

Would you like more details or have any further questions? Here are 5 related questions to explore this topic further:

1. What happens if the interest is compounded quarterly instead of semiannually?
2. How does the future value change if the interest rate increases to 4%?
3. What is the total interest earned over the 5 years?
4. How does the required deposit change if the investment period is extended to 10 years?
5. What would the deposit amount be if the future value desired is $20,000 instead of $16,000?

**Tip:** When interest is compounded more frequently, you earn more total interest over time.

An investment company pays 3% compounded semiannually. You want to have $16,000 in the future. (A) How much should you deposit now to have that amount 5 years from now?

Find out how much you need to deposit today to reach $16,000 in 5 years with a 3% semiannual interest rate. This detailed explanation covers the compound interest formula and exponential growth concepts, ideal for Grades 9-12.

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