Questions: At age 18, someone sets up an IRA (individual retirement account) with an APR of 6%. At the end of each month he deposits 60 in the account. How much will the IRA contain when he retires at age 65? Compare that amount to the total deposits made over the time period. After retirement the IRA will contain . (Do not round until the final answer. Then round to the nearest cent as needed.)

At age 18, someone sets up an IRA (individual retirement account) with an APR of 6%. At the end of each month he deposits 60 in the account. How much will the IRA contain when he retires at age 65? Compare that amount to the total deposits made over the time period.

After retirement the IRA will contain . (Do not round until the final answer. Then round to the nearest cent as needed.)

Transcript text: At age 18, someone sets up an IRA (individual retirement account) with an APR of $6 \%$. At the end of each month he deposits $\$ 60$ in the account. How much will the IRA contain when he retires at age 65 ? Compare that amount to the total deposits made over the time period. After retirement the IRA will contain \$ $\square$ . (Do not round until the final answer. Then round to the nearest cent as needed.)

Solution

Solution Steps

Step 1: Calculate the Future Value of the IRA

The future value $ FV $ of the IRA can be calculated using the formula for the future value of an annuity:

\[ FV = P \cdot \left( \frac{(1 + r)^{nt} - 1}{r} \right) \]

where:

$ P = 60 $ (monthly deposit),
$ r = \frac{0.06}{12} = 0.005 $ (monthly interest rate),
$ nt = 564 $ (total number of deposits).

Substituting the values:

\[ FV = 60 \cdot \left( \frac{(1 + 0.005)^{564} - 1}{0.005} \right) \approx 187912.5582 \]

Thus, rounding to the nearest cent, we have:

\[ FV \approx 187912.56 \]

Step 2: Calculate Total Deposits Made

The total deposits made over the time period can be calculated as:

\[ \text{Total Deposits} = P \cdot nt = 60 \cdot 564 = 33840 \]

Step 3: Calculate Interest Earned

The interest earned can be calculated by subtracting the total deposits from the future value:

\[ \text{Interest Earned} = FV - \text{Total Deposits} = 187912.5582 - 33840 = 154072.5582 \]

Final Answer

The future value of the IRA when the individual retires at age 65 is approximately $ \$187912.56 $. The total deposits made over the time period amount to $ \$33840.00 $. The interest earned during this period is approximately $ \$154072.56 $.

Thus, the final answer is:

\[ \boxed{187912.56} \]

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