Question

at what rate converted every 3 months will a sum of money triple itself in 18 years and 6 months.

Answer 1

I assume you mean compounded

r = interest rate specified usually per year /4 because compounding every quarter year

18.5 years * 4 = 74 periods

3 = x^74
log 3 = 74 log x
log x = .00644758
x =1.015
so
1.5 percent per period or r = 4*1.5 = 6%

Answer 2

or, using the compound interest formula

(1 + i)^74 = 3 , where i is the quarterly rate
take 74th root of both sides
1+i = 1.01495..
i = .01495..
annual rate compounded quarterly
= 4(.01595..)
= .0598 or 5.98%