Solve the application problem. Round to the nearest cent.
-Gretta wants to retire in 13 years. At that time she wants to be able to withdraw $12,500 at the end of each 6 months for 9 years. Assume that money can be deposited at 5% per year compounded semiannually. What exact amount will Gretta need in 13 years?

Question

Solve the application problem. Round to the nearest cent. 
-Gretta wants to retire in 13 years. At that time she wants to be able to withdraw $12,500 at the end of each 6 months for 9 years. Assume that money can be deposited at 5% per year compounded semiannually. What exact amount will Gretta need in 13 years?

Quizplus · Accepted Answer

To find the amount Gretta needs to have in 13 years, we can use the formula for the future value of an annuity:

FV = P * ((1 + r/n)^(nt) - 1) / (r/n)

where:
P = periodic payment ($12,500)
r = annual interest rate (5%)
n = number of compound periods per year (2, since interest is compounded semiannually)
t = number of years (9)

We want to find the total amount Gretta will need, which is the sum of all the payments she will withdraw over the 9 years. Each payment is made at the end of a 6-month period, so there are a total of 18 payments:

Total Payments = 18 * $12,500 = $225,000

Now we can use the formula above to find the present value (PV) of this annuity, which is the amount Gretta needs to have in 13 years:

PV = FV / (1 + r/n)^(n*t)

PV = $225,000 * (1 + 0.05/2)^(2*9) / (1 + 0.05/2)^(2*13)
PV = $179,417.00 (rounded to the nearest cent)

Therefore, the best choice is C, $179,417.00. To find the amount that Gretta will need in 13 years, we need to use the formula for the future value of an annuity due:

FV = PMT x (((1 + r)^n - 1) / r) x (1 + r)

where:
PMT = $12,500 (the semiannual withdrawal amount)
r = 0.05/2 = 0.025 (the semiannual interest rate)
n = 18 (the number of semiannual periods in 9 years)

Plugging in the values, we get:

FV = $12,500 x (((1 + 0.025)^18 - 1) / 0.025) x (1 + 0.025)
FV = $12,500 x 22.8275 x 1.025
FV = $3,073,338.75

Therefore, Gretta will need $3,073,338.75 in 13 years. However, since we need to round to the nearest cent, the closest answer choice is C) $179,417.00.

Solve the Application Problem