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 solve this problem, we will use the **future value of an ordinary annuity formula**, which is appropriate because Mrs. Garcia-Shapiro will be making regular semiannual deposits into an account earning interest. The future value of an ordinary annuity formula is:

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

Where:
- $FV$ = future value of the annuity, which is $120,500
- $P$ = semiannual deposit (this is what we need to find)
- $r$ = interest rate per period (semiannual interest rate)
- $n$ = total number of periods (semiannual periods)

### Step 1: Determine the values for the variables.
- Interest rate per period, $r$: The annual interest rate is 3.83%, and since the interest is compounded semiannually, the semiannual interest rate is:

\[
r = \frac{3.83\%}{2} = \frac{0.0383}{2} = 0.01915
\]

- Number of periods, $n$: Since the investment is for 12 years and the deposits are made semiannually, there are:

\[
n = 12 \times 2 = 24 \, \text{semiannual periods}
\]

- Future value, $FV$ = $120,500

### Step 2: Plug values into the formula and solve for $P$.

\[
120,500 = P \times \frac{{(1 + 0.01915)^{24} - 1}}{0.01915}
\]

Now, let's calculate $P$.Mrs. Garcia-Shapiro would need to deposit approximately **$4,002.21** semiannually into the account to have $120,500 after 12 years, assuming the account earns 3.83% interest compounded semiannually.

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Find out how to calculate the semiannual deposit needed to accumulate $120,500 in 12 years with 3.83% interest compounded semiannually. This solution uses the future value of an ordinary annuity formula and is ideal for high school students studying financial mathematics.

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