Given:
- The sum amounts to Rs. 5150 in 3 years.
- The sum amounts to Rs. 5350 in 7 years.
- The interest is simple interest.

To Find:
The rate of interest.

Assumption:
Let the principal amount be P and the rate of interest be R%.

Approach:
We can solve this problem by setting up two equations using the given information and then solving them simultaneously to find the rate of interest.

Equation 1:
Using the formula for simple interest, we can write the first equation as:
5150 = P + (P * R * 3/100)

Equation 2:
Similarly, using the same formula, we can write the second equation as:
5350 = P + (P * R * 7/100)

Solving the Equations:
We can solve these two equations simultaneously to find the value of R.

Step 1: Multiply both sides of Equation 1 by 100 to eliminate the fraction:
515000 = 100P + 3PR

Step 2: Multiply both sides of Equation 2 by 100 to eliminate the fraction:
535000 = 100P + 7PR

Step 3: Subtract Equation 1 from Equation 2:
(535000 - 515000) = (100P + 7PR) - (100P + 3PR)

Step 4: Simplify the equation:
20000 = 4PR

Step 5: Divide both sides of the equation by 4P:
20000 / 4P = R

Step 6: Simplify the equation:
5000 / P = R

Conclusion:
Therefore, the rate of interest is 5000 / P.