Michael invested $1600 per half-year in an IRA each half-year for 9 years earning 9% compounded semi-annually.

At the end of 9 years he ceased the IRA payments, but continued to invest his accumulated amount at 10% compounded yearly for the next 8 years.
a) What was the value of his IRA at the end of 9 years?
Answer = $
b) What was the value of the investment at the end of the next 8 years?
Answer = $

To calculate the value of Michael's IRA at the end of 9 years, we need to use the formula for compound interest:

A = P(1 + r/n)^(nt)

where:
A = the future value of the investment
P = the principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years

a) Let's calculate the value of Michael's IRA at the end of 9 years:

P = $1600 (invested per half-year)
r = 9% = 0.09
n = 2 (since the interest is compounded semi-annually)
t = 9 years

Since Michael invested $1600 per half-year, we need to multiply the principal amount by the number of compounding periods per year (2):

P_total = P * 2 = $1600 * 2 = $3200

Now we can use the compound interest formula to calculate the future value:

A = P_total(1 + r/n)^(nt)
A = $3200(1 + 0.09/2)^(2 * 9)
A ≈ $3200(1 + 0.045)^18
A ≈ $3200(1.045)^18
A ≈ $6181.458

Therefore, the value of Michael's IRA at the end of 9 years is approximately $6,181.46.

b) Now let's calculate the value of the investment at the end of the next 8 years:

Since Michael ceased the IRA payments and continued to invest his accumulated amount at 10% compounded yearly, we can use the same compound interest formula. However, now we have a different interest rate, compounding period, and time period:

P = $6,181.46 (value of his IRA at the end of 9 years)
r = 10% = 0.10
n = 1 (compounded yearly)
t = 8 years

Using the compound interest formula:

A = P(1 + r/n)^(nt)
A = $6,181.46(1 + 0.10/1)^(1 * 8)
A ≈ $6,181.46(1 + 0.10)^8
A ≈ $6,181.46(1.10)^8
A ≈ $11,390.445

Therefore, the value of the investment at the end of the next 8 years is approximately $11,390.45.

To find the value of Michael's IRA at the end of 9 years, we can use the formula for compound interest:

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

Where:
A = the final amount
P = the principal amount (initial investment)
r = annual interest rate (in decimal form)
n = number of times the interest is compounded per year
t = number of years

a) To find the value of Michael's IRA at the end of 9 years:

P = $1600 (since he invests this amount every half-year)
r = 9% or 0.09 (compounded semi-annually)
n = 2 (compounded semi-annually)
t = 9 years

First, we need to calculate the total number of periods for this investment:

Since he invests every half-year, the total number of periods in 9 years would be twice the number of years, which is 18 periods.

Using the formula:

A = $1600 * (1 + 0.09/2)^(2 * 18)
A = $1600 * (1 + 0.045)^(36)
A = $1600 * (1.045)^(36)
A ≈ $1600 * 2.273103
A ≈ $3636.97

Therefore, the value of his IRA at the end of 9 years is approximately $3636.97.

b) To find the value of the investment at the end of the next 8 years:

Since Michael ceased the IRA payments and continued to invest the accumulated amount at 10% compounded yearly, we can calculate the future value using the same formula:

P = $3636.97 (the previously calculated value of the IRA after 9 years)
r = 10% or 0.1 (compounded yearly)
n = 1 (compounded yearly)
t = 8 years

A = $3636.97 * (1 + 0.1/1)^(1 * 8)
A = $3636.97 * (1 + 0.1)^(8)
A ≈ $3636.97 * 2.143588
A ≈ $7792.59

Therefore, the value of Michael's investment at the end of the next 8 years is approximately $7792.59.