To solve this problem, we need to calculate the future value (FV) of an investment that compounds at different intervals. Let's break it down step-by-step:

Part 1: Future Value with Annual Compounding

The formula to calculate the future value (FV) of an investment with compound interest is:

$FV = PV \times (1 + \frac{r}{n})^{nt}$

where:

• $PV$ is the present value (initial investment),
• $r$ is the annual interest rate,
• $n$ is the number of compounding periods per year,
• $t$ is the number of years.

For this problem:

• $PV = \frac{20,000}{2} = 10,000$ (since only half of the money is invested),
• $r = 10\% = 0.10$,
• $n = 1$ (compounded annually),
• $t = 25$ years.

Plugging these values into the formula:

$FV = 10,000 \times (1 + \frac{0.10}{1})^{1 \times 25}$

Let's calculate it.The future value (FV) of the investment with annual compounding is approximately $108,347.06.

Part 2: Impact of More Frequent Compounding

Now, let's determine if the future value would be larger or smaller if the compounding period was 6 months (semi-annual compounding).

For this, we need to understand the effect of compounding frequency:

• Answer B is correct: "Larger. The greater the number of compounding periods per year, the larger the impact of compound interest, all else equal."

This is because more frequent compounding results in interest being calculated on the accumulated interest more often, leading to a larger future value.

Part 3: Future Value with Semi-Annual Compounding

For semi-annual compounding, we have:

• $n = 2$ (since it compounds twice a year).

Using the same formula but adjusting for semi-annual compounding:

$FV = 10,000 \times \left(1 + \frac{0.10}{2}\right)^{2 \times 25}$

Let's calculate the future value with semi-annual compounding.The future value (FV) of the investment with semi-annual compounding is approximately $114,674.00.

Part 4: Additional Amount Earned with Semi-Annual Compounding

To find the additional amount earned due to the more frequent compounding period, we calculate the difference between the future values of semi-annual and annual compounding:

$\text{Additional Amount} = FV_{\text{semi-annual}} - FV_{\text{annual}}$

Let's compute this difference.The additional amount earned with semi-annual compounding compared to annual compounding is approximately $6,326.94.

Summary

1. Future Value with Annual Compounding: $108,347.06
2. Effect of More Frequent Compounding: Future value is larger (Answer B).
3. Future Value with Semi-Annual Compounding: $114,674.00
4. Additional Amount Earned with Semi-Annual Compounding: $6,326.94

Would you like more details on any part of this solution, or do you have any questions?

Related Questions:

1. How would the future value change if the interest rate were 8% instead of 10%?
2. What would be the future value if the compounding were done quarterly instead of semi-annually?
3. How does continuous compounding compare to annual and semi-annual compounding in terms of future value?
4. If the compounding period were reduced to monthly, how much more would be earned compared to semi-annual compounding?
5. What is the effect of different investment durations (e.g., 10 years, 15 years) on the future value with semi-annual compounding?

Tip:

When dealing with compound interest problems, remember that more frequent compounding leads to higher future values, provided the interest rate and principal amount remain the same.