Interest is compounded quarterly at 3.5 percent. If $1300 is invested, how many years will it take to triple?

3 = 1(1 + .035/4)^4t

3 = 1.00875^4t

ln 3 / ln 1.00875^4 = t

31.526049672939399010999896954013 = t

31.53 years

If it's 3.5% per year compounded quarterly, then it's 3.5%/4 = 0.875% per quarter. So each quarter you have 1 + (0.875/100) = 1 + 0.00875 = 1.00875 times the balance from the previous quarter. So after a year you have 1.00875^4 = 1.03546206054931640625 times the balance from the previous year. So we have:

1.03546206054931640625^x = 3

Since x is a real number:

x = log[1.03546206054931640625](3)

x =~ 31.526049672939399010999896954013 years

Note that 31.5 years is not enough time, and after that you don't get paid more interest until the end of the 3rd quarter of the 32nd year, so we have to round UP to the nearest quarter, so it becomes 31.75.