The difference between compound interest and simple interest for 2 yr at rate of 5% per annum is ? 5, then find the sum.

Answer

To solve the problem of finding the sum (P) when the difference between compound interest (CI) and simple interest (SI) for 2 years at a rate of 5% per annum is 5, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Rate (R) = 5% per annum
- Time (T) = 2 years
- Difference between CI and SI = 5

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

3. **Calculate Compound Interest (CI):**
The formula for Compound Interest is:
\[ CI = A - P \]
where \( A \) is the amount after 2 years, calculated as:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the values:
\[ A = P \left(1 + \frac{5}{100}\right)^2 = P \left(1.05\right)^2 = P \times \frac{21}{20} \]
Therefore, the Compound Interest (CI) is:
\[ CI = P \times \frac{21}{20} - P = P \left(\frac{21}{20} - 1\right) = P \left(\frac{21 - 20}{20}\right) = \frac{P}{20} \]

4. **Set up the equation for the difference:**
The difference between CI and SI is given as:
\[ CI - SI = 5 \]
Substituting the expressions for CI and SI:
\[ \frac{P}{20} - \frac{P}{10} = 5 \]

5. **Combine the fractions:**
To combine the fractions, find a common denominator:
\[ \frac{P}{20} - \frac{2P}{20} = 5 \]
This simplifies to:
\[ -\frac{P}{20} = 5 \]

6. **Solve for P:**
Multiply both sides by -20:
\[ P = -5 \times -20 = 100 \]

### Final Answer:
The sum (P) is **100**.