On what sum does the difference between the compound interest and the simple interest for 3 years at 10% is Rs. 31?

Answer

To solve the problem, we need to find the principal amount (P) for which the difference between the compound interest (CI) and simple interest (SI) for 3 years at a rate of 10% is Rs. 31.

### Step-by-Step Solution:

1. **Understand the Formula for the Difference between CI and SI**:
The difference between compound interest and simple interest for 3 years can be calculated using the formula:
\[ \text{Difference} = P \times \frac{R}{100} \times \frac{R}{100} \times 3 \]
where \( P \) is the principal amount, and \( R \) is the rate of interest.

2. **Substitute the Values**:
Here, \( R = 10\% \) and the difference is given as Rs. 31. Plugging in the values, we get:
\[ 31 = P \times \frac{10}{100} \times \frac{10}{100} \times 3 \]

3. **Simplify the Equation**:
Simplifying the right side:
\[ 31 = P \times \frac{10}{100} \times \frac{10}{100} \times 3 = P \times \frac{10 \times 10 \times 3}{100 \times 100} = P \times \frac{300}{10000} \]
This simplifies to:
\[ 31 = P \times \frac{3}{100} \]

4. **Isolate P**:
To find \( P \), we can rearrange the equation:
\[ P = 31 \times \frac{100}{3} \]

5. **Calculate P**:
Now, calculate the value of \( P \):
\[ P = \frac{3100}{3} \approx 1033.33 \]

6. **Final Result**:
Therefore, the principal amount \( P \) is approximately Rs. 1033.33.