The difference of S.I. and C.I on a sum at 10% per annum for 3 years is Rs. 31 Find the sum.

Answer

To solve the problem of finding the principal amount given the difference between Simple Interest (SI) and Compound Interest (CI) over 3 years at a rate of 10% per annum, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for Simple Interest (SI)**:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

2. **Calculate the Simple Interest for 3 years**:
Given \( R = 10\% \) and \( T = 3 \) years, the formula becomes:
\[ SI = \frac{P \times 10 \times 3}{100} = \frac{30P}{100} = 0.3P \]

3. **Understand the formula for Compound Interest (CI)**:
The formula for CI after \( T \) years is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where \( A \) is the total amount after \( T \) years.

4. **Calculate the Compound Interest for 3 years**:
Using \( R = 10\% \) and \( T = 3 \):
\[ A = P \left(1 + \frac{10}{100}\right)^3 = P \left(1.1\right)^3 \]
Calculating \( (1.1)^3 \):
\[ (1.1)^3 = 1.331 \]
Thus,
\[ A = 1.331P \]
The Compound Interest (CI) is then:
\[ CI = A - P = 1.331P - P = 0.331P \]

5. **Find the difference between CI and SI**:
The difference between CI and SI is given by:
\[ CI - SI = 0.331P - 0.3P = 0.031P \]
According to the problem, this difference is Rs. 31:
\[ 0.031P = 31 \]

6. **Solve for the principal \( P \)**:
To find \( P \), divide both sides by 0.031:
\[ P = \frac{31}{0.031} = 1000 \]

### Final Answer:
The principal amount is Rs. 1000.

---