Suppose you have decided to retire as soon as you have saved $\mathbf{\$ 1 3 0 , 0 0 0}$. Your plan is to put $\$ 300$ each month into an ordinary annuity that pays an annuity interest rate of $\mathbf{8} \%$. If you are now $\mathbf{2 0}$ years old, how old will you be when you retire?
years
Done

Step 1 ：The problem is asking for the time it will take to save a certain amount of money by depositing a fixed amount into an annuity each month. The annuity has a certain interest rate.

Step 2 ：The formula for the future value of an ordinary annuity is: \(FV = P * [(1 + r/n)^(nt) - 1] / (r/n)\) where: \(FV\) is the future value, \(P\) is the monthly deposit, \(r\) is the annual interest rate, \(n\) is the number of times that interest is compounded per year, and \(t\) is the number of years.

Step 3 ：We can rearrange this formula to solve for \(t\): \(t = \log((FV * r/n + P) / P) / (n * \log(1 + r/n))\)

Step 4 ：We know that \(FV = \$130,000\), \(P = \$300\), \(r = 8\% = 0.08\), and \(n = 12\) (since interest is compounded monthly). We can plug these values into the formula to find \(t\).

Step 5 ：Then, we add the current age to \(t\) to find the retirement age.

Step 6 ：The calculated retirement age is approximately 37.03 years. However, since we cannot have a fraction of a year in age, we need to round this up to the nearest whole number. This is because even if you reach the required savings a few months after your birthday, you will still be considered a year older in terms of age.

Step 7 ：Final Answer: The person will be \(\boxed{38}\) years old when they retire.