Suppose Mrs. Garcia-Shapiro wanted to have $120,500 in her account after 12 years, How much would she need to deposit semiannually into an account earning 3.83% interest compounded semiannually?

To determine how much Mrs. Garcia-Shapiro needs to deposit semiannually to reach $120,500 in 12 years, we'll use the formula for the future value of an annuity compounded periodically:

\[
FV = P \times \frac{(1 + r/n)^{nt} - 1}{r/n}
\]

Where:
- $ FV $ is the future value, which is $120,500.
- $ P $ is the amount of each semiannual deposit (what we're solving for).
- $ r $ is the annual interest rate (3.83% = 0.0383 as a decimal).
- $ n $ is the number of compounding periods per year (since it's compounded semiannually, $ n = 2 $).
- $ t $ is the total number of years (12 years).

Now, we substitute these values into the formula:

\[
120,500 = P \times \frac{(1 + \frac{0.0383}{2})^{2 \times 12} - 1}{\frac{0.0383}{2}}
\]

Let's solve this step by step to find $ P $, the semiannual deposit amount.It seems like I can’t do more advanced data analysis right now. Please try again later.

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Here are five related questions to expand on the topic:
1. What is the difference between simple interest and compound interest?
2. How does the frequency of compounding affect the total amount of interest earned?
3. Can you calculate the future value if the interest were compounded quarterly instead of semiannually?
4. How would this scenario change if Mrs. Garcia-Shapiro made annual deposits instead of semiannual ones?
5. What happens to the required deposit if the interest rate increases?

**Tip:** The more frequently an account compounds interest, the more total interest you'll earn over time.

Discover how to calculate the semiannual deposit needed to reach $120,500 in 12 years, with a 3.83% interest rate compounded semiannually. This detailed solution breaks down the future value of annuities formula, guiding students through compound interest and semiannual payments. Ideal for high school math (Grades 11-12).

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