The difference between simple and compound interest on a certain sum for 2 years at 5% per annum compounded annually is ₹ 75. Find the sum:

Options

A. ₹ `40000`
B. ₹ `36000`
C. ₹ `45000`
D. ₹ `30000`

Answer

To solve the problem, we need to find the principal amount (sum) given the difference between compound interest (CI) and simple interest (SI) for 2 years at a rate of 5% per annum is ₹75.

### Step-by-Step Solution:

1. **Define the Principal Amount**:
Let the principal amount be \( P \).

2. **Calculate Compound Interest (CI)**:
The formula for compound interest for 2 years at a rate of 5% compounded annually is:
\[ CI = P \left(1 + \frac{r}{100}\right)^n - P \]
where \( r = 5\% \) and \( n = 2 \).
\[ CI = P \left(1 + \frac{5}{100}\right)^2 - P \]
\[ CI = P \left(1.05\right)^2 - P \]
\[ CI = P \left(1.1025\right) - P \]
\[ CI = P \times 1.1025 - P = P \times 0.1025 \]

3. **Calculate Simple Interest (SI)**:
The formula for simple interest is:
\[ SI = \frac{P \times r \times t}{100} \]
where \( r = 5\% \) and \( t = 2 \).
\[ SI = \frac{P \times 5 \times 2}{100} = \frac{10P}{100} = 0.1P \]

4. **Find the Difference Between CI and SI**:
The difference between compound interest and simple interest is:
\[ CI - SI = P \times 0.1025 - 0.1P \]
\[ = P \times (0.1025 - 0.1) = P \times 0.0025 \]

5. **Set the Difference Equal to ₹75**:
According to the problem, this difference is ₹75:
\[ P \times 0.0025 = 75 \]

6. **Solve for P**:
To find \( P \), divide both sides by 0.0025:
\[ P = \frac{75}{0.0025} \]
\[ P = 30000 \]

### Final Answer:
The principal amount is ₹30,000.

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