A sum of money is invested at 10% per annum compounded half-yearly. If the difference of amounts at the end of 6 months and 12 months is Rs 189, find the sum of money invested,

Answer

To solve the problem step by step, we will follow the formula for compound interest and apply it to find the sum of money invested.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the sum of money invested be \( P \) (in rupees).
- The rate of interest is \( 10\% \) per annum.
- The interest is compounded half-yearly.

2. **Calculate the Amount after 6 Months**:
- The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
- Here, \( r = 10\% = 10 \), \( n = 2 \) (since it's compounded half-yearly), and \( t = \frac{1}{2} \) (for 6 months).
- Substitute the values into the formula:
\[ A_{6\text{ months}} = P \left(1 + \frac{10}{2 \times 100}\right)^{2 \times \frac{1}{2}} = P \left(1 + \frac{10}{200}\right)^{1} = P \left(1 + 0.05\right) = P \times 1.05 \]
- Therefore, the amount after 6 months is:
\[ A_{6\text{ months}} = 1.05P \]

3. **Calculate the Amount after 12 Months**:
- Now, for 12 months, \( t = 1 \) year.
- Substitute the values into the formula:
\[ A_{12\text{ months}} = P \left(1 + \frac{10}{2 \times 100}\right)^{2 \times 1} = P \left(1 + 0.05\right)^{2} = P \left(1.05\right)^{2} \]
- Calculate \( (1.05)^{2} \):
\[ (1.05)^{2} = 1.1025 \]
- Therefore, the amount after 12 months is:
\[ A_{12\text{ months}} = 1.1025P \]

4. **Set Up the Equation for the Difference**:
- According to the problem, the difference between the amounts at the end of 6 months and 12 months is Rs 189:
\[ A_{12\text{ months}} - A_{6\text{ months}} = 189 \]
- Substitute the amounts:
\[ 1.1025P - 1.05P = 189 \]
- Simplify the equation:
\[ (1.1025 - 1.05)P = 189 \]
\[ 0.0525P = 189 \]

5. **Solve for \( P \)**:
- Divide both sides by \( 0.0525 \):
\[ P = \frac{189}{0.0525} \]
- Calculate \( P \):
\[ P = 3600 \]

6. **Conclusion**:
- The sum of money invested is Rs 3600.

### Final Answer:
The sum of money invested is Rs 3600.

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