**Given Information:**
- Mahesh borrowed a certain sum for 2 years at simple interest from Bhim Mahesh.
- Mahesh borrowed the same sum for 2 years at compound interest from Vishnu.
- At the end of 2 years, Mahesh paid rupees 400 as simple interest.
- At the end of 2 years, Mahesh paid rupees 410 as compound interest.

**To Find:**
- The sum borrowed by Mahesh.
- The rate of interest.

**Solution:**

Let's assume that the principal amount borrowed by Mahesh is 'P' and the rate of interest is 'R'.

**Simple Interest:**

The formula for calculating simple interest is: S.I. = (P * R * T) / 100
where P is the principal amount, R is the rate of interest, and T is the time period.

Given that Mahesh paid rupees 400 as simple interest for 2 years, we can write the equation as:

400 = (P * R * 2) / 100

Simplifying the equation, we get:

2PR = 40000 (Equation 1)

**Compound Interest:**

The formula for calculating compound interest is: C.I. = P(1 + R/100)^T - P
where P is the principal amount, R is the rate of interest, and T is the time period.

Given that Mahesh paid rupees 410 as compound interest for 2 years, we can write the equation as:

410 = P(1 + R/100)^2 - P

Simplifying the equation, we get:

(1 + R/100)^2 = 410/P + 1 (Equation 2)

**Solving Equations 1 and 2:**

We can solve equations 1 and 2 simultaneously to find the value of P and R.

From Equation 1, we have:

2PR = 40000

From Equation 2, we have:

(1 + R/100)^2 = 410/P + 1

Simplifying the second equation, we have:

(1 + R/100)^2 - 1 = 410/P

(1 + R/100)^2 = (410 + P)/P

Taking the square root on both sides, we get:

1 + R/100 = sqrt((410 + P)/P)

R/100 = sqrt((410 + P)/P) - 1

Simplifying further, we have:

R = 100(sqrt((410 + P)/P) - 1)

Now, we can substitute the value of R in Equation 1 to find the value of P.

2P * 100(sqrt((410 + P)/P) - 1) = 40000

Simplifying the equation, we can solve it to find the value of P.

Once we find the value of P, we can substitute it in Equation 1 or Equation 2 to find the value of R.

Therefore, the sum borrowed by Mahesh and the rate of interest can be determined by solving the above equations.

We know that SI = PTR/100

                         P = SI * 100/r*t

                            = 400 * 100 / r * 2

                            = 20000/R 

We know that CI = P(1+r/100)^n - P
 
                            = P(1+r/100)^n - 1

                     410 = 20000/R(1+r/100)^n - 1

                     410 = 20000/R(10000 + R^2 + 2000R - 10000)/10000

                     410 = 2/R(R^2 + 200R)

                     410R = 2(R^2+200R)

                     2R^2 + 400R - 410R = 0

                     2R^2 = 10R

                    Rs =5

P = 20000/5

   = 4000.


Sum = 4000, Rate =5%