Find the compound interest on Rs 1000 at the rate of 10% per annum for 18 months when interest is com- pounded half-yearly.

Answer

To find the compound interest on Rs 1000 at the rate of 10% per annum for 18 months when the interest is compounded half-yearly, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Principal (P)**:
- The principal amount (P) is given as Rs 1000.

2. **Convert the Rate of Interest (R)**:
- The annual interest rate is 10%. Since the interest is compounded half-yearly, we need to divide the rate by 2.
- \( R = \frac{10\%}{2} = 5\% \)

3. **Convert the Time Period (T)**:
- The time period is given as 18 months. Since we are compounding half-yearly, we need to convert this into half-years.
- \( T = \frac{18 \text{ months}}{6 \text{ months}} = 3 \text{ half-years} \)

4. **Use the Compound Interest Formula**:
- The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
- Substituting the values:
\[ A = 1000 \left(1 + \frac{5}{100}\right)^3 \]
- Simplifying inside the parentheses:
\[ A = 1000 \left(1 + 0.05\right)^3 = 1000 \left(1.05\right)^3 \]

5. **Calculate \( (1.05)^3 \)**:
- \( (1.05)^3 = 1.157625 \) (You can calculate this using a calculator or by multiplying \( 1.05 \times 1.05 \times 1.05 \))

6. **Calculate the Amount (A)**:
- Now, substituting back:
\[ A = 1000 \times 1.157625 = 1157.625 \]

7. **Calculate the Compound Interest (CI)**:
- The compound interest is given by:
\[ CI = A - P \]
- Substituting the values:
\[ CI = 1157.625 - 1000 = 157.625 \]

### Final Answer:
The compound interest on Rs 1000 at the rate of 10% per annum for 18 months when compounded half-yearly is **Rs 157.625**.

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