(b) If the employee invests $$ \$ 200 $$ instead of $$ \$ 100 $$ at $$ 4.6 \% $$, find the value of the annuity after 18 yr . Round the answer to the nearest cent. Compare the result to part (a). The value of the annuity after 18 yr is approximately $$ \$ $$ Doubling the monthly contribution approximately (Choose one) T the value of the annuity.

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To find the value of the annuity after 18 years with a monthly investment of $$ \$200 $$ at an interest rate of $$ 4.6\% $$, we can use the future value of an annuity formula:

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

Where:
- $$ FV $$ is the future value of the annuity.
- $$ P $$ is the monthly contribution.
- $$ r $$ is the monthly interest rate (annual rate divided by 12).
- $$ n $$ is the total number of contributions (number of years multiplied by 12).

### Given:
- Monthly contribution $$ P = 200 $$
- Annual interest rate = $$ 4.6\% = 0.046 $$
- Monthly interest rate $$ r = \frac{0.046}{12} \approx 0.00383333 $$
- Number of years = 18
- Total contributions $$ n = 18 \times 12 = 216 $$

### Calculation:
1. Calculate the future value:

$$
FV = 200 \times \frac{(1 + 0.00383333)^{216} - 1}{0.00383333}
$$

2. Calculate $$ (1 + 0.00383333)^{216} $$:

$$
(1 + 0.00383333)^{216} \approx 2.030856
$$

3. Now substitute back into the formula:

$$
FV = 200 \times \frac{2.030856 - 1}{0.00383333}
$$

$$
FV = 200 \times \frac{1.030856}{0.00383333} \approx 200 \times 268.000
$$

$$
FV \approx 53600.00
$$

### Final Result:
The value of the annuity after 18 years is approximately $$ \$53,600.00 $$.

### Comparison:
If in part (a) the monthly contribution was $$ \$100 $$, we can calculate that value as well for comparison.

Using the same formula with $$ P = 100 $$:

$$
FV = 100 \times \frac{(1 + 0.00383333)^{216} - 1}{0.00383333}
$$

$$
FV = 100 \times \frac{1.030856}{0.00383333} \approx 100 \times 268.000 \approx 26800.00
$$

### Conclusion:
- The value of the annuity after 18 years with a monthly contribution of $$ \$200 $$ is approximately $$ \$53,600.00 $$.
- The value of the annuity with a monthly contribution of $$ \$100 $$ is approximately $$ \$26,800.00 $$.

### Doubling the Monthly Contribution:
Doubling the monthly contribution approximately **doubles** the value of the annuity. In this case, $$ \$200 $$ is double $$ \$100 $$, and the future value reflects that increase.
The value of the annuity after 18 years with $$ \$200 $$ monthly contribution is approximately $$ \$53,600.00 $$. Doubling the monthly contribution approximately doubles the value of the annuity.

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