On what sum of money will the difference between simple interest and compound interest for 2 years at 5% per annum be equal to rupes 50?

Answer

To solve the problem, we need to find the principal amount (sum of money) for which the difference between the simple interest (SI) and compound interest (CI) after 2 years at a rate of 5% per annum is equal to ₹50.

### Step-by-Step Solution:

1. **Define Variables:**
Let the principal amount be \( P \) (or \( x \)). The rate of interest \( R \) is 5% per annum, and the time period \( T \) is 2 years.

2. **Formulas:**
- The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
- The formula for Compound Interest (CI) after 2 years is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

3. **Calculate Simple Interest:**
Using the values:
\[ SI = \frac{P \times 5 \times 2}{100} = \frac{10P}{100} = \frac{P}{10} \]

4. **Calculate Compound Interest:**
First, calculate \( \left(1 + \frac{5}{100}\right)^2 \):
\[ \left(1 + \frac{5}{100}\right) = 1.05 \]
\[ CI = P \left(1.05^2\right) - P = P(1.1025) - P = 0.1025P \]

5. **Find the Difference:**
The difference between CI and SI is:
\[ CI - SI = 0.1025P - \frac{P}{10} \]
Convert \( \frac{P}{10} \) to a decimal:
\[ \frac{P}{10} = 0.1P \]
Therefore:
\[ CI - SI = 0.1025P - 0.1P = 0.0025P \]

6. **Set the Difference Equal to ₹50:**
According to the problem, this difference equals ₹50:
\[ 0.0025P = 50 \]

7. **Solve for \( P \):**
\[ P = \frac{50}{0.0025} = 20000 \]

### Final Answer:
The principal amount (sum of money) is ₹20,000.

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