The difference between compound interest for a year payable half-yearly and simple interest on a certain sum of money lent out at 10% for a year is Rs 15. Find the sum of money lent out.

Answer

To solve the problem step by step, we will follow the process of calculating both the simple interest and the compound interest, and then find the principal amount based on the given difference.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Rate of interest (r) = 10% per annum
- Time (t) = 1 year
- Difference between compound interest (CI) and simple interest (SI) = Rs 15

2. **Calculate Simple Interest (SI):**
- The formula for Simple Interest is:
\[ SI = \frac{P \times r \times t}{100} \]
- Substituting the values:
\[ SI = \frac{P \times 10 \times 1}{100} = \frac{P}{10} \]

3. **Calculate Compound Interest (CI) for Half-Yearly Compounding:**
- For half-yearly compounding, the rate is halved and the time is doubled:
- Half-yearly rate = \( \frac{10}{2} = 5\% \)
- Time in half-years = \( 1 \times 2 = 2 \) half-years
- The formula for Compound Interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
- Substituting the values:
\[ A = P \left(1 + \frac{5}{100}\right)^2 = P \left(1 + 0.05\right)^2 = P \left(1.05\right)^2 \]
- Calculating \( (1.05)^2 \):
\[ (1.05)^2 = 1.1025 \]
- Therefore, the amount (A) becomes:
\[ A = P \times 1.1025 \]
- The Compound Interest (CI) is:
\[ CI = A - P = P \times 1.1025 - P = P(1.1025 - 1) = P \times 0.1025 \]

4. **Set Up the Equation Using the Given Difference:**
- We know that the difference between CI and SI is Rs 15:
\[ CI - SI = 15 \]
- Substituting the expressions for CI and SI:
\[ P \times 0.1025 - \frac{P}{10} = 15 \]

5. **Simplify the Equation:**
- Convert \( \frac{P}{10} \) to a decimal:
\[ \frac{P}{10} = 0.1P \]
- The equation becomes:
\[ 0.1025P - 0.1P = 15 \]
- Simplifying further:
\[ (0.1025 - 0.1)P = 15 \]
\[ 0.0025P = 15 \]

6. **Solve for P:**
- Divide both sides by 0.0025:
\[ P = \frac{15}{0.0025} = 6000 \]

7. **Final Answer:**
- The sum of money lent out is Rs 6000.