The difference between the compound interest and simple interest. on a certain sum at 3% per annum for 3 years is Rs. 27.27. The sum is :

Answer

To find the sum (P) for which the difference between the compound interest (CI) and simple interest (SI) at a rate of 3% per annum for 3 years is Rs. 27.27, we can follow these steps:

### Step 1: Define the Variables
Let the principal amount (sum) be \( P \).
The rate of interest \( R = 3\% \).
The time period \( T = 3 \) years.

### Step 2: Calculate Simple Interest (SI)
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI = \frac{P \times 3 \times 3}{100} = \frac{9P}{100} \]

### Step 3: Calculate Compound Interest (CI)
To find the Compound Interest for 3 years at 3%, we first need to determine the effective rate. The formula for the effective rate over multiple years is:
\[ \text{Effective Rate} = A + B + C + \frac{A \times B + B \times C + C \times A}{100} + \frac{A \times B \times C}{100} \]
Here, \( A = B = C = 3\% \):
\[ \text{Effective Rate} = 3 + 3 + 3 + \frac{3 \times 3 + 3 \times 3 + 3 \times 3}{100} + \frac{3 \times 3 \times 3}{100} \]
Calculating:
\[ = 9 + \frac{9 + 9 + 9}{100} + \frac{27}{100} = 9 + \frac{27}{100} + \frac{27}{100} = 9 + 0.27 = 9.27\% \]

Now, we can calculate the Compound Interest using the effective rate:
\[ CI = \frac{\text{Effective Rate} \times P}{100} = \frac{9.27P}{100} \]

### Step 4: Set Up the Equation
According to the problem, the difference between CI and SI is Rs. 27.27:
\[ CI - SI = 27.27 \]
Substituting the values we calculated:
\[ \frac{9.27P}{100} - \frac{9P}{100} = 27.27 \]
This simplifies to:
\[ \frac{(9.27 - 9)P}{100} = 27.27 \]
\[ \frac{0.27P}{100} = 27.27 \]

### Step 5: Solve for P
Multiply both sides by 100:
\[ 0.27P = 2727 \]
Now, divide both sides by 0.27:
\[ P = \frac{2727}{0.27} = 10000 \]

### Final Answer
The sum \( P \) is Rs. 10,000.

---