To find the rate of interest, we can use the formula for compound interest:

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

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

We have two equations based on the given information:

Equation 1: A1 = P(1 + r/n)^(nt) = P(1 + r/n)^(2n) = P(1 + r/n)^2 = 600
Equation 2: A2 = P(1 + r/n)^(nt) = P(1 + r/n)^(3n) = P(1 + r/n)^3 = 938

We can divide Equation 2 by Equation 1 to eliminate the variable P:

A2/A1 = (P(1 + r/n)^3) / (P(1 + r/n)^2)
A2/A1 = (1 + r/n)^(3n - 2n)
A2/A1 = (1 + r/n)^n

We know that A2/A1 = 938/600 = 1.5633

Now, we need to find the value of (1 + r/n)^n that satisfies this equation. We can use trial and error method or approximation techniques.

Here, we'll use the trial and error method to find a suitable value for (1 + r/n)^n.

1. Assume a value for n (e.g., 2, 4, 12, etc.) and calculate (1 + r/n)^n.
2. Compare the calculated value with 1.5633.
3. Adjust the assumed value of n accordingly and repeat steps 1-2 until a value is found that is close to 1.5633.

Let's assume n = 2:
(1 + r/2)^2 = 1.5633
(1 + r/2) = √1.5633
1 + r/2 = 1.2498
r/2 = 1.2498 - 1
r/2 = 0.2498
r = 0.2498 * 2
r = 0.4996

Therefore, the rate of interest is approximately 0.4996 or 49.96%.

(Note: This is an example solution using the trial and error method. The actual solution may require further approximation techniques or the use of a calculator.)