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This problem requires calculating the total interest earned by Charles over a 24-year period, considering a monthly investment, a break in contributions after 16 years, and a quarterly compounded interest rate.

Here's the breakdown of the steps involved:

1. **Monthly Contributions**:
   - Charles contributes $300 at the beginning of each month for 16 years.
   - Total contributions period: 16 years = $16 \times 12 = 192$ months.

2. **Investment Growth**:
   - After 16 years of contributing, Charles stops making payments, but the accumulated amount continues to grow with compound interest for another 8 years (from year 16 to year 24).

3. **Interest Rate**:
   - The interest rate is 4.3% compounded quarterly. 
   - Quarterly compounding means we divide the annual rate by 4 and use the compounded formula with periods counted in quarters.

### Step 1: Calculate the Future Value of an Annuity for the First 16 Years

For the first 16 years, Charles is making monthly deposits. We use the future value of an annuity formula for these regular payments:

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

where:
- $ P = 300 $ (monthly payment),
- $ r = \frac{0.043}{4} \approx 0.01075 $ (quarterly interest rate),
- $ n = 16 \times 4 = 64 $ quarters.

Since contributions are monthly, and the compounding is quarterly, we can either adjust the formula to account for monthly deposits with quarterly compounding or find an approximate effective quarterly rate for monthly compounding.

Let me calculate the final result.The total interest Charles earned over the 24-year period is approximately **$58,208.38**.

Would you like more details on any of the steps or have further questions?

### Related Questions:
1. How would the interest earned change if Charles contributed for 20 years instead of 16?
2. What would happen to the final amount if the interest rate was 5% instead of 4.3%?
3. How does quarterly compounding compare to monthly compounding for this scenario?
4. What if Charles stopped contributions after only 10 years—how would the interest amount differ?
5. How can you calculate the effective interest rate if compounding frequency changes?

#### Tip:
When calculating compound interest with regular contributions, ensure the compounding frequency matches the deposit interval for accuracy.

Charles contributed $300 into an investment at the beginning of each month for 16 years, with an interest rate of 4.3% compounded quarterly. How much interest did Charles earn over the 24-year period after he stops making payments?

Learn how to calculate the total interest earned on a monthly investment of $300 compounded quarterly at 4.3% over 24 years. Charles makes contributions for 16 years and stops, allowing the investment to grow for an additional 8 years. This problem involves future value of annuities and compound interest calculations.

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