Starting at age 50 , a woman puts $\$ 1400$ at the end of each quarter into a retirement account that pays $7 \%$ interest compounded quarterly. When she reaches age 60 , she withdraws the entire amount and places it in a mutual fund account that pays $9 \%$ compounded monthly. From then on she deposits $\$ 500$ in the same mutual fund at the end of each month. How much is in the account when she reaches age $65 ?$
At age 65 she has $\$ \square$.
(Simplify your answer. Type an integer or a decimal. Round intermediate steps to the nearest cent. Round the final answer to the nearest dollar if needed.)
Answer
Rounding to the nearest dollar, we get \(\boxed{117840}\).
Step 1 ：First, we need to calculate the amount in the retirement account when the woman reaches age 60. She deposits $1400 at the end of each quarter for 10 years with an interest rate of 7% compounded quarterly.
Step 2 ：We can use the formula for the future value of an ordinary annuity to calculate the amount in the account. The formula is: \(FV = P \times \left[(1 + \frac{r}{n})^{nt} - 1\right] / \left(\frac{r}{n}\right)\), where \(FV\) is the future value of the annuity, \(P\) is the amount of each payment, \(r\) is the annual interest rate (in decimal form), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested for in years.
Step 3 ：Substituting the given values into the formula, we get \(FV1 = 1400 \times \left[(1 + \frac{0.07}{4})^{4 \times 10} - 1\right] / \left(\frac{0.07}{4}\right)\), which simplifies to \(FV1 = 80127.7874548829\).
Step 4 ：Next, we need to calculate the amount in the mutual fund account when the woman reaches age 65. She deposits the entire amount from the retirement account into the mutual fund account, and then deposits $500 at the end of each month for 5 years with an interest rate of 9% compounded monthly.
Step 5 ：Again, we use the formula for the future value of an ordinary annuity. Substituting the given values into the formula, we get \(FV2 = 500 \times \left[(1 + \frac{0.09}{12})^{12 \times 5} - 1\right] / \left(\frac{0.09}{12}\right)\), which simplifies to \(FV2 = 37712.06846277138\).
Step 6 ：Finally, we add the two amounts to get the total amount in the account when the woman reaches age 65. This gives us \(total = 80127.7874548829 + 37712.06846277138 = 117839.85591765428\).
Step 7 ：Rounding to the nearest dollar, we get \(\boxed{117840}\).
