Mrs. Engleder has made deposits of $750.00 at the end of every six months for twenty-three years. If interest is 3.72% compounded monthly, how much will Mrs. Engleder have accumulated six years after the last deposit?
Question 4 options:

$2765.41

$179,523.57

$67,483.03

$243,500.54

This is a problem of calculating the future value of an annuity (the series of deposits made by Mrs. Engleder), followed by compound interest accumulation after the final deposit.

### Step 1: Calculate the Future Value of the Annuity

The formula for the future value of an ordinary annuity (since the deposits are made at the end of each period) is:

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

Where:
- $FV$ is the future value of the annuity.
- $P$ is the periodic payment (in this case, $750.00).
- $r$ is the periodic interest rate.
- $n$ is the total number of periods.

**Given:**
- Annual interest rate = 3.72% compounded monthly, so the monthly interest rate $r = \frac{3.72\%}{12} = \frac{0.0372}{12} = 0.0031$ (approx).
- Payments are made every six months, so we need to convert the rate to a semi-annual equivalent: 
\[
r_{semi} = (1 + 0.0031)^{6} - 1
\]
- Number of years = 23 years, and since payments are made semi-annually, the total number of periods $n = 23 \times 2 = 46$.

### Step 2: Value After Six Years of Final Deposit

After the final deposit, the accumulated amount will grow for another six years with compound interest. The new balance after 6 years can be calculated with the compound interest formula:

\[
A = P \times (1 + r)^n
\]

Where:
- $A$ is the future value after 6 years.
- $P$ is the value of the annuity at the end of 23 years.
- $r$ is the monthly interest rate (compounded monthly).
- $n$ is the number of months for the additional 6 years, $n = 6 \times 12 = 72$.

Let me compute the final values.Mrs. Engleder will have accumulated approximately **$67,483.11** six years after her last deposit.

This matches the third option: **$67,483.03**.

Would you like further details or explanations? Here are some related questions for deeper understanding:

1. How would the result change if the interest rate was compounded quarterly instead of monthly?
2. How would a different payment schedule (e.g., quarterly instead of semi-annually) affect the final amount?
3. What if Mrs. Engleder made her deposits at the beginning of each period instead of at the end?
4. How does the length of the compounding period (monthly, yearly) impact future value?
5. What effect does a higher interest rate have on the future value of the annuity?

**Tip:** The longer the money stays invested after the last deposit, the more it benefits from compounding interest.

Mrs. Engleder has made deposits of $750.00 at the end of every six months for twenty-three years. If interest is 3.72% compounded monthly, how much will Mrs. Engleder have accumulated six years after the last deposit?

Learn how to calculate the future value of an annuity with compound interest, using Mrs. Engleder's deposits as an example. Understand key financial formulas like the future value of an ordinary annuity and compound interest to find how much she will accumulate after six years of her last deposit.

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