Twenty years ago Gabe invested $2,000. For the first ten years he earned 13% compounded semi-annually. For the next ten years he earned 8% compounded quarterly. What was the value of the investment now, at the end of the 20 years?

2000 * (1+.13/2)^(2*10) * (1+.08/4)^(4*10) = 15,560.70

To find the value of the investment at the end of 20 years, we need to calculate the future value of the investment based on the given interest rates.

First, let's calculate the first ten years of the investment. The formula to calculate the future value of an investment with compound interest is:

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

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

For the first ten years, Gabe earned 13% interest compounded semi-annually. So, the values we will use are:

P = $2,000
r = 13% = 0.13
n = 2 (semi-annual compoundings)
t = 10

Plug in these values into the formula:

A = 2000(1 + 0.13/2)^(2*10)
A = 2000(1 + 0.065)^(20)
A = 2000(1.065)^(20)

Now, calculate the value of A using a calculator or a spreadsheet:

A = $2,000 * (1.065)^(20)
A ≈ $6,100.66

Initially, after the first ten years, the investment would be worth approximately $6,100.66.

Next, let's calculate the next ten years of the investment. Gabe earned 8% interest compounded quarterly during this period. Using the same formula with the updated values:

P = $6,100.66
r = 8% = 0.08
n = 4 (quarterly compoundings)
t = 10

Plug in these values into the formula:

A = 6100.66(1 + 0.08/4)^(4*10)
A = 6100.66(1 + 0.02)^(40)
A = 6100.66(1.02)^(40)

Again, calculate the value of A using a calculator or a spreadsheet:

A = $6,100.66 * (1.02)^(40)
A ≈ $14,749.73

Therefore, at the end of the 20 years, the value of the investment would be approximately $14,749.73.