Introduction:
Compound interest is a type of interest that is calculated on the initial principal amount as well as on the accumulated interest of previous periods. In this problem, we have to find the rate of interest for two different time periods based on the given compound interest amounts.

Given:
Compound interest for two years = Rs. 600
Compound interest for three years = Rs. 938

Formula:
The formula for calculating compound interest is:

A = P (1 + R/100)^t

Where A is the amount, P is the principal, R is the rate of interest, and t is the time period.

Solution:

Let us assume the principal amount is x.

For two years:
A = P (1 + R/100)^t
600 = x (1 + R/100)^2
(1 + R/100)^2 = 600/x
1 + R/100 = √(600/x)
R/100 = √(600/x) - 1
R = 100(√(600/x) - 1)

For three years:
A = P (1 + R/100)^t
938 = x (1 + R/100)^3
(1 + R/100)^3 = 938/x
1 + R/100 = ∛(938/x)
R/100 = ∛(938/x) - 1
R = 100(∛(938/x) - 1)

Substituting:
Now, we can substitute the value of R in the above equations and solve for x.
We get,
R = 100(√(600/x) - 1)
R = 100(∛(938/x) - 1)

Equating both equations, we get,

100(√(600/x) - 1) = 100(∛(938/x) - 1)
√(600/x) - 1 = ∛(938/x) - 1
√(600/x) = ∛(938/x)
(600/x)^(1/2) = (938/x)^(1/3)
(600/x)^(3/2) = (938/x)
x = 600*938/((938)^(2/3)*600^(1/2))

Substituting the value of x in any of the above equations, we can calculate the rate of interest.

Conclusion:
The rate of interest for two years is (100(√(600/x) - 1))% and for three years is (100(∛(938/x) - 1))%.