To calculate the future value of Sam's IRA account, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial money).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of times that interest is compounded per year.
- \(t\) is the number of years the money is invested or borrowed for.
For Sam's account:
- \(P = 4000\) (the initial deposit)
- \(r = 0.098\) (the annual interest rate of 9.8% expressed as a decimal)
- \(n = 12\) (since the interest is compounded monthly)
- \(t = 8\) (the time from age 25 to 33, which is 8 years)
Now, let's plug in the values into the formula:
\[ A = 4000 \left(1 + \frac{0.098}{12}\right)^{12 \times 8} \]
Calculating it step by step:
- Calculate \(\frac{0.098}{12}\):
\[ \frac{0.098}{12} \approx 0.00816667 \]
- Calculate \(1 + \frac{0.098}{12}\):
\[ 1 + 0.00816667 \approx 1.00816667 \]
- Calculate \(12 \times 8\):
\[ 12 \times 8 = 96 \]
- Now compute \(\left(1.00816667\right)^{96}\):
\[ (1.00816667)^{96} \approx 2.225 \]
- Finally, calculate \(A\):
\[ A \approx 4000 \times 2.225 \approx 8900 \]
Thus, the value of Sam's IRA when he is 33 years old will be approximately:
\[ \text{The account will be worth about } $8900. \]