# The compound interest on a certain sum for 2 years is Rs 41 and simple interest is Rs 40.Then the rate per annum is

## To find the rate per annum, we can use the formulas for compound interest and simple interest.

Compound interest is calculated using the formula:
A = P (1 + r/n)^(n*t)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment/loan amount (the initial amount)
r = annual interest rate (in decimal form)
n = number of times that interest is compounded per year
t = number of years

Simple interest is calculated using the formula:
A = P + (P * r * t)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment/loan amount (the initial amount)
r = annual interest rate (in decimal form)
t = number of years

Given that the compound interest is Rs 41 and the simple interest is Rs 40 for 2 years, we can set up the equations:

P (1 + r/n)^(n*t) = P + (P * r * t)

Let's assume the principal amount P is Rs X.

For the compound interest:
X (1 + r/n)^(n*t) = X + (X * r * t)
X (1 + r/n)^(n*2) = X + (X * r * 2)

For the simple interest:
X + (X * r * 2) = X + 40

Now, we need to solve these two equations to find the rate per annum (r). We can simplify the equations as follows:

Equation 1:
X (1 + r/n)^(n*2) = X + (X * r * 2)
Divide both sides by X:
(1 + r/n)^(n*2) = 1 + r*2

Equation 2:
X*r*2 = 40
Divide both sides by X*2:
r = 20/X

Now, substitute r = 20/X in Equation 1:
(1 + 20/(X*n))^2n = 1 + 20

To solve for the rate per annum (r), we need to assign a value to the principal amount (X) and find the value of n that satisfies the equation. Unfortunately, without additional information, it is not possible to determine the exact value of the rate per annum.