The difference between compound and simple interest on a sum of money deposited for 2 years at 5% per annum is rupes 12 . Find the sum of money.

Answer

To solve the problem, we need to find the sum of money (principal) based on the difference between compound interest (CI) and simple interest (SI) over 2 years at a rate of 5% per annum, which is given as ₹12.

### Step-by-Step Solution:

1. **Define Variables**:
Let the principal amount (sum of money) be \( P = x \).
The time \( t = 2 \) years and the rate \( r = 5\% \).

2. **Formulate the Simple Interest (SI)**:
The formula for Simple Interest is:
\[ SI = \frac{P \times r \times t}{100} \]
Substituting the values:
\[ SI = \frac{x \times 5 \times 2}{100} = \frac{10x}{100} = \frac{x}{10} \]

3. **Formulate the Compound Interest (CI)**:
The formula for Compound Interest is:
\[ CI = P \left(1 + \frac{r}{100}\right)^t - P \]
Substituting the values:
\[ CI = x \left(1 + \frac{5}{100}\right)^2 - x \]
Simplifying:
\[ CI = x \left(1 + 0.05\right)^2 - x = x \left(1.05\right)^2 - x \]
Calculating \( (1.05)^2 \):
\[ (1.05)^2 = 1.1025 \]
Therefore:
\[ CI = x \times 1.1025 - x = 1.1025x - x = 0.1025x \]

4. **Set Up the Equation**:
According to the problem, the difference between CI and SI is ₹12:
\[ CI - SI = 12 \]
Substituting the expressions for CI and SI:
\[ 0.1025x - \frac{x}{10} = 12 \]

5. **Convert SI to a Common Denominator**:
The term \( \frac{x}{10} \) can be converted to have a denominator of 100:
\[ \frac{x}{10} = \frac{10x}{100} \]
Thus, we rewrite the equation:
\[ 0.1025x - 0.1x = 12 \]

6. **Combine Like Terms**:
Converting \( 0.1025x \) to a fraction:
\[ 0.1025x = \frac{1025x}{10000} \]
And \( 0.1x = \frac{1000x}{10000} \):
\[ \frac{1025x - 1000x}{10000} = 12 \]
This simplifies to:
\[ \frac{25x}{10000} = 12 \]

7. **Solve for \( x \)**:
Multiply both sides by 10000:
\[ 25x = 120000 \]
Divide by 25:
\[ x = \frac{120000}{25} = 4800 \]

8. **Final Answer**:
The sum of money (principal) is ₹4800.