To solve for the future value of Ryan's investment using compound interest, we’ll use the formula:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

where:

• $A$ = the future value of the investment (the amount we are solving for),
• $P$ = the principal investment amount (initial deposit), which is $5,900,
• $r$ = the annual interest rate (decimal), which is 5% or 0.05,
• $n$ = the number of times that interest is compounded per year, which is monthly (so $n = 12$),
• $t$ = the number of years the money is invested, which is 11 years.

Step-by-Step Calculation:

1. Substitute the values into the formula:

$A = 5900 \left(1 + \frac{0.05}{12}\right)^{12 \times 11}$

2. Calculate the monthly interest rate:

$\frac{0.05}{12} = 0.0041667$

3. Multiply the compounding periods per year by the number of years:

$12 \times 11 = 132$

4. Substitute back to get:

$A = 5900 \left(1 + 0.0041667\right)^{132}$

5. Calculate $(1 + 0.0041667)^{132}$:

$A = 5900 \times (1.0041667)^{132}$

6. After calculating the power, you’ll find:

$A \approx 5900 \times 1.744$

7. Multiply to find the future value:

$A \approx 10289.60$

So, the amount in the account after 11 years will be approximately $10,289.60.

Would you like further details, or have any questions?

Related Questions:

1. What would the future value be if the compounding was done quarterly instead of monthly?
2. How does changing the interest rate affect the future value after 11 years?
3. What if Ryan invested an additional amount each year? How would that alter the calculations?
4. Can we calculate the total interest earned over the 11 years?
5. How would the answer change if Ryan withdrew a fixed amount annually?

Tip:

For accurate results, always remember to round intermediate values only at the final step to avoid rounding errors.