a. The couple needs to invest approximately 619.46 at the end of each 6-month period for the next 18 years to begin making their college withdrawals.
b. Since they can't make a partial withdrawal, they will be able to make 4 college withdrawals before the account balance is 0.
a. To calculate how much the couple needs to invest at the end of each 6-month period for the next 18 years, we can use the formula for the future value of an annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = future value
P = periodic payment
r = interest rate per period
n = number of periods
In this case, FV is 30,000, P is what we need to find, r is 8% compounded semiannually (or 4% per 6-month period), and n is 8 years (or 16 periods).30,000 = P * ((1 + 0.04)^16 - 1) / 0.04
30,000 = P * (1.04^16 - 1) / 0.0430,000 = P * (2.9376 - 1) / 0.04
30,000 = P * 1.9376 / 0.0430,000 = P * 48.44
P = 30,000 / 48.44
P ≈619.46
So, the couple needs to invest approximately 619.46 at the end of each 6-month period for the next 18 years to begin making their college withdrawals.
b. With the inheritance of 38,000 contributed to the account after 8 years, we can calculate the new future value of the annuity using the formula:
FV = P * ((1 + r)^n - 1) / r + A * (1 + r)^n
Where:
A = inheritance amount
FV = future value
P = periodic payment
r = interest rate per period
n = number of periods
Using the new future value and the regular withdrawal amount of 30,000, we can calculate how many college withdrawals they will be able to make before the account balance is0.
First, we need to find the new future value after the inheritance:
FV = 619.46 * ((1 + 0.04)^16 - 1) / 0.04 +38,000 * (1 + 0.04)^16
FV ≈ 619.46 * 48.44 +38,000 * 2.9376
FV ≈ 30,000 +111,657.28
FV ≈ 141,657.28
Now we can calculate the number of withdrawals:
Number of withdrawals = FV / withdrawal amount
Number of withdrawals ≈141,657.28 / 30,000
Number of withdrawals ≈ 4.72
Since they can't make a partial withdrawal, they will be able to make 4 college withdrawals before the account balance is 0.