Given:
- The sum of money amounts to Rs. 575 after 3 years.
- The sum of money amounts to Rs. 625 after 5 years.
- The rate of interest is the same for both periods.

To find:
- The rate of interest per annum.

Explanation:

Let's assume that the principal amount is P and the rate of interest is R% per annum.

Step 1: Calculating the Simple Interest for the first case:
- Principal amount (P) after 3 years = Rs. 575
- Simple Interest (SI) = Principal amount (P) * Rate of interest (R) * Time (T)
- SI = P * R * T
- SI = 575 - P (As the principal amount is given)
- 575 - P = P * R * 3 (Equation 1)

Step 2: Calculating the Simple Interest for the second case:
- Principal amount (P) after 5 years = Rs. 625
- Simple Interest (SI) = Principal amount (P) * Rate of interest (R) * Time (T)
- SI = P * R * T
- SI = 625 - P (As the principal amount is given)
- 625 - P = P * R * 5 (Equation 2)

Step 3: Solving the equations:
- From equation 1, we have 575 - P = 3PR
- From equation 2, we have 625 - P = 5PR

Step 4: Finding the value of P:
- Subtracting equation 1 from equation 2, we get:
625 - P - (575 - P) = 5PR - 3PR
50 = 2PR
- Dividing both sides by 2R, we get:
P = 25/R

Step 5: Substituting the value of P in equation 1:
575 - (25/R) = 3R
Multiplying both sides by R, we get:
575R - 25 = 3R^2
3R^2 - 575R + 25 = 0

Step 6: Solving the quadratic equation:
Using the quadratic formula, we get:
R = (-b ± √(b^2 - 4ac)) / 2a

Where a = 3, b = -575, and c = 25.

By substituting the values in the formula, we get two possible values for R:
- R ≈ 5.15
- R ≈ 2.03

Step 7: Selecting the appropriate rate of interest:
Since the rate of interest cannot be negative, we take the positive value of R. Therefore, the rate of interest is approximately 5.15% per annum.

Conclusion:
The rate of interest per annum is approximately 5.15%.