https://file.aiquickdraw.com/mathbot/file/17287936024258qif4zbx.jpg

To solve this problem, we need to use the formula for the present value of a compound interest investment:

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

Where:
- $ PV $ = Present Value (the amount you should deposit now),
- $ FV $ = Future Value (the amount you want to have in the future, which is $12,000),
- $ r $ = Annual interest rate (7%, or 0.07),
- $ n $ = Number of times the interest is compounded per year (semiannually means $ n = 2 $),
- $ t $ = Number of years (5 years in this case).

### Step-by-step solution:
1. **Given values:**
   - $ FV = 12,000 $,
   - $ r = 0.07 $,
   - $ n = 2 $,
   - $ t = 5 $.

2. **Substitute these into the formula:**

\[
PV = \frac{12,000}{\left(1 + \frac{0.07}{2}\right)^{2 \times 5}}
\]

3. **Simplify the equation:**
\[
PV = \frac{12,000}{\left(1 + 0.035\right)^{10}} = \frac{12,000}{(1.035)^{10}}
\]

4. **Calculate $ (1.035)^{10} $:**
\[
(1.035)^{10} \approx 1.4106
\]

5. **Calculate the present value (PV):**
\[
PV = \frac{12,000}{1.4106} \approx 8,505.39
\]

So, you should deposit approximately **$8,505.39** now to have $12,000 in 5 years.

Would you like more details on any step? 

Here are some related questions:
1. How does the frequency of compounding affect the amount you need to deposit?
2. What happens if the interest rate changes to 6% instead of 7%?
3. How would the formula change if the interest is compounded quarterly?
4. How does the present value formula differ from the future value formula?
5. What is the effect of increasing the investment period from 5 years to 10 years?

**Tip:** Always double-check whether interest is compounded annually, semiannually, or at another interval—it greatly affects the result!

An investment company pays 7% compounded semiannually. You want to have $12,000 in the future. How much should you deposit now to have that amount 5 years from now? (Round to the nearest cent.)

Learn how to calculate the present value of an investment when compounded semiannually at 7%. This problem explores the compound interest formula to determine how much you should deposit now to have $12,000 in 5 years. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation